This video presents advanced mental math techniques for the Mental Math World Cup Grandmaster category (ages 12+), covering cube roots (using last digit analysis and first digit estimation), square roots (with methods to resolve two possible last digits), multiplication (cross multiplication method), division (last digit analysis and algorithm), and HCF/LCM (Euclid's algorithm and the HCF-LCM relationship). The key insight is that mental math at elite levels relies on understanding the psychology of calculation and strategic digit analysis rather than memorizing formulas.
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MMWC 2026 Webinar Grandmaster Category & Zen Master Challenge
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Good.
>> Good afternoon, [clears throat] sir.
>> Good afternoon.
>> Good evening, sir.
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>> Good afternoon, sir.
>> Happy Sunday.
>> Thank you.
>> Thank you, sir. What should I say?
>> Good afternoon.
>> Good afternoon, sir.
>> Good afternoon.
One more.
>> Thank you. Good afternoon.
It's a good it's so good to see all of you here today this Sunday you know I I I don't know if I should say Sunday afternoon Sunday morning Sunday evening everywhere around the world good afternoon or you know happy Sunday is a better thing to say um I'm Ainas Shetty from live math competitions and leagues and we are here today for the for the metal math tips and tricks session for the grandmaster category that's ages 12 and above for the mental math world cup 2026. I'm sure all of you are very excited about participating in in the mental math world cup. It's something that is going to be you know around your mind for for some time now. So welcome all of you.
Although I'm sure you have a lot of questions about about the event, how is it going to be held, you know, what, when, where, all of this information has been shared with you on your registered email ID. Once you register, the emails that was sent to you included all these details about the competition along with links to join the WhatsApp community groups and uh the study kit. So if you still have any more questions about the competition or about the study kit or about the uh community link, please uh write to us on contact at greatlivemcl.com or drop us a message on WhatsApp on +98291027238 and somebody from our team will definitely answer you and respond to all your questions about the competition.
Today's session is not about um you know uh the competition. It's about mental math tips and tricks and for that we have someone very special Mr. Daniel Tims who is a worldrenowned mental math trainer as well as expert and coach specializing in practical mental math and psychology of calculation. He's actively involved in providing workshops, training stage performances and training material for mental math including international competitions. Daniel also runs the world mental calculation website offering news and resources for advanced and competitive mental math. Apart from that, he also competes in mental math competitions around the world. He's amongst the fastest you know calendar date calculators for for UK. So it's a privilege to have him here with us today and especially here to for all of you. So make the most out of this session. I hope you have a pen and paper ready over here. We are not here to just you know we are not here to Daniel is not here to teach you the basics of mental mathematics. Uh he he will show you the psychology of it. How the way to think about a problem and how to go about solving it you know. So normal school math is obviously something that you've learned. You've learned your abacus and all of that. But there are still other ways to look to think about how to solve these problems and Daniel is here for that. I would appreciate if you focus on that. Make sure you have a pen and paper ready and make the most document everything. Obviously, the recording of this session will also be available with you tomorrow. So, you know, if if you still happen to miss something, you can you can catch up on on it again later. Daniel, with that said, I think you have the stage now.
>> Oh, thank you very much Aanash for the introduction and very nice to see all of you joining the workshop today from all around the world. I'm curious where people are from. Um, if I invite you to send a message in the chat just to tell me which country you are joining us from today.
Um, me for example, I'm from England.
I'm joining from Oxford in the UK where I spend the summer.
And just to expand on the introduction that Avanash has kindly shared. So yeah, this is the um workshop for ticks and tricks and tips for mental maths specifically for the things that you need for the Grandmaster and Zen master levels of the mental math world cup that's coming up. I'm Daniel Tims. I kind of introduction to me. I'm yeah I'm from the UK. I did mathematics at university. I worked as a software developer from 2012 after I finished university and then I decided I was going to travel. So since 2016 I have been moving between places quite a lot, staying in several different countries each year.
And one of the reasons I started was because I was going to a competition in Las Vegas and I thought I don't want to go there only for this. I want to make a bigger trip and uh and visit other places. So I um yeah, so mental maths has also been a part of my travels. I've been several places around the world to compete in mental maths and you'll be aware that life competitions and league.
They offer competitions online as well as in different places around Asia.
And so in the last years I have concentrated a lot on mental maths. I have the mental calculation world uh world's mental calculation website that some of you may have seen. It's like to use patient recognition instead of repeatedly show >> admin can you please mute everyone >> according to the >> thank you. Uh yeah, so I have the world calculation website which is where I publish uh a lot of resources about mental maths in general, especially advanced methods and sometimes interviews with calculators, things like this. And I'm also a mental math coach at events such as this one, which is why I'm here today. Today we're going to cover uh exact square roots, multiplications, divisions, um addition and subtraction. here thinking more about the psychology and not the methods because these are quite basic highest common factor and lowest common multiple and starting with exact cube roots. So these are the main topics that I'm going to cover today and multiplication and division. This also includes percentages.
I've got lots of messages from people about where they are in the world. So where do we have we have from India from Delhi we have more from India from Romania from India from the UAE from Bangaluru South Africa more from the UAE other cities in India. Many people from India India is always very strong for mental maths. A lot of people from Dubai as well or other places in the UAE from Bahrain.
more people from Bahrain, Fujayra, I think that's in the UAE, Sri Lanka. I was in Sri Lanka in October actually. I was staying a week in Colbo and more people from SA South Africa or I think Saudi um could be either >> Saudi Arabia. Yeah, >> Saudi Arabia. Yeah. Okay. more people from Sri Lanka. So mental math you see is a way of when you are at a a high level at mental math which is what you're all working towards now then you get uh access to a more international community where you meet people from all around the world from Europe from Asia from Africa from the Americas who are also elite at mental maths. So, in the competitions that I've been to in person, again, there's people from all sorts of places around the world. Here online today, we have people from Mauritius, from Thailand. I was in Chiang Mai, not Bangkok, but I was in Chiang Mai for a few months at the end of last year. Very nice place. Germany, I've been to Germany many times for mental um calculation competitions.
[snorts] Uh it's another place where a lot of competitions have been organized.
More people from India and finally more people from India. So okay, really nice to have you all joining us here today. I see the emoji reactions with lots of flags as well. Very cool.
So what are we doing today? I said we're going to start with cube roots.
So I'm going to share the whiteboards and we get started.
>> Hey Daniel before you start u a request to everyone u you know if you today's session will be interactive Daniel will be asking you a lot of questions while he's going through all the topics but when you respond to it please respond with one response you know if you give a answer just put it once and just leave it at that. Don't keep on putting that same answer again and again but that because that you know clogs the entire chat and we are not able to call on any anybody else after that. All right. So it's a simple request. Please don't you know block the chats.
>> Thank you. Yeah. We had the issue yesterday. Some people were sending the message um and response many times and it actually means I don't find their answer at all because um when I have a look at the chat later I can't understand the message. So just send a message one time and then I will try to include as many of you as I can during the questions today.
So we're looking today first of all at cube roots. So what does a cube root mean? So if I tell you for example that 15 * 15 * 15 is equal to 3,375.
Another way of saying this is that 15 cubed is 3375.
So this means the cube root of 3375.
Let's have that back is 15. So in the mental math world cup and also other calculation competitions you may be asked to find the cube root of a specific number and for the ones that you have in the mental math world cup you know that the answer is going to be a whole number. So you if you are asked to find the cube root of 3375 you know that the answer is going to be a whole number and not a decimal.
So, how do we do this? First of all, I'm going to share the cube numbers up to 9 cubed.
And can anyone tell me answers in the private chat? What do you notice about the last digits of all of these? [snorts] So, answers in the private chat. And when I for this workshop when you send me the answer to a question in the private chat sometimes I will ask for a volunteer to explain. If you would like to volunteer to tell us your answer then after you write the correct answer then tell me that you volunteer and then I will select someone at random who has the correct answer and who would like to volunteer. This way, if you want to answer but don't want to volunteer, then I'm not going to surprise you.
So, I will look and see if there's someone with the correct answer who would like to volunt who says that they would like to volunteer. I've got some correct answers, but no volunteers so far.
one question.
>> Um, okay.
>> Yeah. Um, how do you chat? I can't find the chat.
>> If I think at the bottom of your screen, if you put the mouse there for one second, then a little bubble will appear. Well, a lot of options will appear and one of them is called chats.
>> Oh. Um, like when I press everyone, it says chat disabled.
>> You can send a message only to me. I'm Daniel Tim.
>> Oh, okay. Okay.
>> Thank you.
>> Yeah, maybe that's helpful for other people. Yeah. So, to send a message to me, then send it to Daniel Tims.
>> Thanks, >> Daniel. Just hold on a minute, right?
>> Yeah. Yeah, we can go. We can move.
Okay, thank you. So remember that if you would like to volunteer to explain your answer to us then tell uh write the word volunteer in the chats. So today the our first volunteer is um Retif from South Africa.
>> So can we unmute Ratif please?
Thank you.
>> You said ratif I E F. Yeah.
>> R T.
>> Morning sir.
>> Good morning. What did you notice about the units digits?
>> I noticed that all of the last digits were all of them were different.
>> Good. Yeah, they're in a kind of random looking order, but they're all different.
Um, exactly. Yeah. Well, so well done for everyone who noticed this. And this is going to help us a lot for calculating the cube roots.
Next question. And this might look like a really difficult question, but um what um what is the last digit of just the last digit?
What is the last digit of this number?
The answer is a really really big number. But what is the last digit?
again write the answer in the chat and if you volunteer to share the answer then you can also write volunteer.
So lots of answers not all of them correct but it's good to see good to see people trying okay can we invite range please range Agrawell uh the the like one thing I noticed which I also wanted to add is that the cube of any number which is ending with seven the answer the unit digit of the answer will always be three. So that should be the answer.
>> Very good. Yeah. So the the answer here to find the last digit of the cube we don't have to look at all of the other digits. We only need to look at the seven. We only need to look at the seven. And the last digit for sure is going to be the same as the cube of seven. So it's the same as for 7 cubed.
Must end in three. So the answer is going to be a huge number, but the last digit for sure is going to be a three.
And so this means so first of all why is this true? Some of you at school have been studying algebra. So I'll show you as a kind of bonus today why this is true.
I guess I need more space for this. So boom boom.
Yeah. So I'll explain just with some algebra why this is true. So all of the algebra you're use you're doing at school what is the purpose of this? Well many many many things and in this case we can show that the rules in mental maths are working. So any number that ends in seven can be written as a multiple of 10 + 7.
So this is a multiple of 10 + 7.
Its cube is therefore going to be 10 n + 7 all cubed.
And if we do the algebra on this, this is maybe a little more advanced algebra, but I tell you the answer. It's going to be 1,00 m cubed plus 700 m^ 2 + 490 m + 343 which is equal to I'm going to copy and paste most of this.
Actually I'm going to paste all of this.
So this is 10 multiplied by all of this + 3. So what I've done here is I've written the cube of any number that ends in seven as a multiple of 10 + 3.
[clears throat] So this means that any the cube of any number in seven we can prove with algebra it ends in a three.
What about the cube of any number that ends in four? Well, we can do the same thing and then um uh we will see that it must end in a four and we can do the same thing for an eight and show that it must end in a two. So we've got two things that we found today. First of all that for the cubes the units this the units digits are all different. And the second thing is that the cube of the number depends only on the last digits.
So logically this means that if I give you a cube number, you can look at the last digits and see all of the possibilities for the last digit in the cube root, but there's only going to be one. So it means we know it for sure. So putting this together, we now have a rule for calculating um cube roots.
So if I have um I just have make an example here.
Um don't know why anyone is looking for Okay.
So for example to find the cube root of this number step one is going to be find last digits which is a three.
see which cube ends in this digit.
So in this case we know that 3 43 is 7 cubed and therefore last digit must be for example I can't type must be seven.
So whenever you have a cube root question the first thing you can do is look at the last digits. In this case, it would be a three. And say which cube number from 1 to 9 ends in a three. And you should know that 7 cubed is 343.
And therefore the answer uh the cube root of 103,823 must end in seven.
So this is the this is the first thing.
So a question for everyone to try this.
What is the last digit of the cube root of 116 bill786,755,000 72?
So answers in the the private chat.
Okay. And for a volunteer, can we have Dash, please? Dash has the correct answer.
Many other people also have the correct answer. Very good.
>> D I'm trying to unmute you. Can you d Muler?
>> Uh yes sir.
The correct answer is eight.
>> Good. Yeah. And why is it eight? How how can we be sure that the answer is eight?
>> The answer is eight because the cube of 8 is 5.2 which ends in the digit two.
Very good. Yeah. So, so the cube ends in two.
It's like 512 which is 8 cubed.
Therefore, cube root ends in 8.
Okay, very nice. So, thank you for sharing. This is exactly correct.
So, this is how we can find the first digit sorry the last digit of any cube number even if it's a really big cube root. And some people they practice doing cube roots of eight and nd-digit numbers or even like 10 11 12digit numbers or sometimes even larger. And this is the first step that they can do.
How do we find the other digit though?
So in the example of 103,823 we we also need to find the first uh the other digits [snorts] in the answer.
So question for everyone.
Can you tell me what is 40 cubed?
What is 40 cubed?
I write the question down so you can also read it.
What is 40 cubed?
Okay. Uh, can we invite Mia to tell us the answer?
Mia, can you tell us what's um what's 40 cubed?
>> Sir, it is 64,000.
As the zeros have to end in three digits and since 4 cubed is 64, it's supposed to be 64,000.
>> Very good. Yeah, thank you. So, uh 40 itself is just 4 * 10. So that means that 40 cubed is equal to 4 cubed * 10 cubed which is equal to 64 * a,000. So 64,000.
So this means that for calculating a cube root, we can look at the last three digits or [snorts] should I say we can ignore the last three digits and [snorts] see um see what the previous three digits are. So in this example, let's use the same color. If I remove the last three digits, which is 8 Q3.
So, I'm going to put a a cross through those last three digits and look only at the 103. I can now compare that in in blue to my list of cubes. And in this case, 103 is greater than 64, but it's less than 125. So, that means that the cube root it must be more than 40, but less than 50. So, it must be 40 something. It's 40 something, but we know it ends in seven. So the answer must be 47.
And so putting these together, we have a complete method for calculating the cube root of any number. And I'll do this on the the next slide of the whiteboard. We can also save the whiteboard and send them to you later.
So if you're making notes now, that's really good. And we will also give you the whiteboard to add to your notes later.
So to find the cube root of uh four, five or sixdigit number and this is what you need to do for the mental math world cup. If it's a seven-digit number or larger then we have to do more things in the methods. But today I want to focus on the tips and tricks for the mental math world cup.
I'll also give you some information later so you can read by yourself how to find cube roots of even larger numbers if you want to study that. So step one we've already said it's look at the last um look at the last digits and see which cube has the same last digit.
That's the last digit of the cube.
step. I'm not going to call it step two.
I'm going to call it step A because you can do step one first or you can do step A first. It doesn't matter. You get two pieces of information and those are the two digits of the answer. So you could do step A first and then do step one. So to illustrate that, I'm going to do I'm going to put it in this order. So step A is remove the last three digits.
Compare these to the cubes of the digits 1 to 9.
That's the first digit of the cube. So this here is now a full method for calculating the cube root of any number.
So in this case we have um 103,823.
So I'll draw this as an example to the side. Which color should we use? Let's do red for example this number.
So step a cube root is between 40 and 50.
So it's 40 something.
Step one ends in three is a bit like 3 43.
Last digit is seven.
Everything together it's 40 something. It ends in a seven.
Therefore, the answer therefore the answer is 47.
I've written this in red in lots of words, but there's when you practice this, it's very very fast to go through these two steps, get the two pieces of information, and then write the answer.
So, let's try this with an example.
I'll give an example that's not too crazy. So, let's try writing in orange. Example one cube root of 592,74.
So it answers in the private chat and if you want to volunteer, you can also tell me that you volunteer.
And just so you know, I I'm getting like a 100 messages. So I it's it's really random whether I see your message or not.
So what's the cube root of this one?
Okay, first correct answer I see is from Ayan Ayan Banso.
Lots of other correct um answers coming through.
>> So should I >> Yeah. What do you think the answer is?
Uh so sir first I took the uh first three digits 5 92 uh now 592 is between uh 8 cube and 9 cube uh so that means that uh the first digit is 8 then I looked at the last digit the last digit was four and I saw uh and only when it's four cube uh is the last digit four so the that means the second digit is four uh then you take 8 and four so 84 4 84 is the answer.
>> Very nice.
So it ends in four like 64 which is 4 cubed. So units digit is 4 and so the answer is 84. Very good and nice explanation.
Thank you for that. Very good. So the correct answer here is 84 and I see in the chat quite a lot of other people also giving me the answer of 84. So if you got 84 then yes well done and lots of volunteers as well. Thank you very much for that.
Okay, let's do one more example of the cube roots.
And um yeah, another example for the cube roots.
So in this case, I'm going to pick I'll do a try a tricky one.
So again, same thing. answers in the private chat and if you want to volunteer then write the word volunteer after your answer.
So cube root of this number Hey Daniel, while solving this sum um if you could also explain uh to the participants for the first digit why would you choose an eight instead of a nine or for example I'm just giving you an example of you know why would you choose the first digit where Which one is closer to the actual answer >> for the for the orange example? Okay.
Yes. So, we know that 8 cubed is 512.
So, this means 80 cubed must be 512,000.
And so, the number we've got is 5, not 12, 592, and something. So, it's larger than 80 cubed. So, we know that it must be 80 something. It could be 81. It could be 88. It could be 89. We don't know. And um but we know that it must be it can't be 90 or more because 90 cubed would be 729,000 and that's bigger than 592,000.
So I'm just seeing which of these cube numbers in blue is the thousand's digits greater than. In this case, it's greater than 512.
Okay, for this I've got a few different answers coming through. And so this is a bit of a tricky question apparently.
I've I'm glad I gave you this one. Can I ask for aentia?
Aentia.
>> Yes, sir.
>> Yeah. So, what method did you use? And finally we get the answer.
>> So sir we like keep the like when we remove the last three digits um we get a we get 12 which is like nearest to 8 the cube of two. So we put so the first digit is 2 and uh then we use uh the last digit of 1 167 which is seven. So 7 can only be a cube of three. So that is why I wrote uh it is cube of 23.
Very good. Yeah, great explanation there. So, a lot of people, they're tempted to look at the first three digits, which is 121, and then you might think it's like 125, but really it should be less or it's more than 64. Uh, but it's not the point. In this case here, we're looking at the first two digits only, which is 12 or 012 if you like. And 12 when we compare it, it's greater than 8. So it must be 20 something because 8 is 2 cubed. The 10's digit must be two. It's not as big as 27. So it can't be 30 or more. And the last digit then is the last digit is what is it? Seven.
Therefore ends in three.
So the answer is going to be 23. So I think that's um trick quite a few people but it's really important to not be tricked by this because in the mental math world cup you will have cube roots of sixdigit but also fivedigit and also fourdigit numbers and so important to get those right because I got so many different answers I want to do one more example of this before moving on. So example in brown then H um I just get the value on my calculator.
Okay.
So what is the cube root of 582?
Sorry. 5832 too.
Okay. Correct answer from Aisha Hammed.
Can we have Aisha Hammed to share please? [snorts] >> Um yes.
Yeah. How did you solve this one? You got the correct answer.
>> First, I did step one. I looked at the last digit and it was a two. And the two came from the number eight.
And then I looked at the first number after removing the last three digits. It was a five. and five was just above the number one.
So the cube number is 18.
>> Very good. Yeah. So when we take the three digits away, we're left only with five. Not a three-digit number, not a two-digit number, just a single digit number. So when we compare this to the cube numbers, then it's between one and eight. It's above one. So between one and eight. So it's above one. So the answer is going to be 10 something.
It's going to be 18.
So also notice something here. Aisha did step one first and in the previous examples we did step A first. Which one is correct? It doesn't matter. you get the two pieces of information in whichever order you want and solve the question. So this is a great illustration actually solving the question using the same method but doing step one before step a. Thank you for sharing.
So um this is everything I want to mention I think about cube roots. [snorts] So, we've done some examples of the standard method that works for any cube number where the answer is two digits. If you want to do cube roots of greater ones, then you have to do some extra steps that makes it more difficult. You don't need that for this particular competition. We also proved mathematically that this rule works, which we don't need to prove it when we're solving, but it's nice to see why this is true.
Okay, so that's everything I want to cover with the cube roots. So now let's move on to the next topic which is going to be the exact square roots.
So exact square roots. So what does it mean? If I have 45 squared, this is 2025.
Uh well this means 45 * 45 which is 2025.
So therefore square t of 2025 is 45. So this is just what it means. Hopefully you all know this. If you didn't then you discovered this today. So it's like cube roots except in cube roots you have the number multiplied by itself three times. But square roots it's like a two-dimension. So, it's only multiplied twice.
So, how do we do this?
Um, well, let's try to do it the same way as the cube roots. So, we can look at the last digits and hope that they're all the same. And we can look at the first digit and figure out the two digits this way. And when we try this, we see that it almost works, but not quite.
So I'll I'll show you how I would try to do the methods for let's say cube uh sorry square roots of 3,364 and then you'll see uh what's going wrong. So step a is I take away the last three digits. No no no no no not last three digits. That's [snorts] not going to work because you see if I have um 50 squared, this is 25,500.
I've only got two zeros at the end from the fact that it's 50, not five.
Um again, I can have 60^ 2. This is equal to 36 with two zeros.
So need to remove last two digits only.
Not the last three digits, the last two digits. So in step a 33 64 I've got 33 and and then uh I removing the last two digits at the end.
Compare 33 to square numbers.
So 25 is greater than or equal to sorry is less than or equal to 33 which is less than 36. Therefore the 10 digit is like 5^2 = 25.
So the answer is 50 something.
So step A is working the same except that I don't take away three digits for a cube root. I just take away two digits for the square roots.
So so far we know the answer is going to be 50s something but we're trying to find out what the last digit is. So the last um let's say step one last digit is four.
This is like 2^2 = 4. Therefore, the answer is 52.
Answers in the private chat. What have I done incorrectly? What is my mistake?
Put a big cross here so you understand this is incorrect. Could I have a volunteer to explain what have I done wrong?
So, put the right. Tell me the correct answer. And and then the word volunteer.
Okay. Uh, can we have yimchan please? Yachimchan.
Um it's uh you are incorrect because uh 8^2 is equal to 64 and 64 also has a four at the end.
>> Good. Yeah, this is exactly right. So um this is like 2^ 2 or it's like 8 squar equ= 64.
So from this information I don't know whether it's 52 or 58 and actually it will be 58.
So for square roots you have the problem that when you look at the last digit there's usually two possibilities and you need to do some extra thinking to decide which of these it's going to be.
So, how to decide whether it's small 52 or large 58. You always have a small possibility or a large possibility.
And I'm going to give you two methods.
So, well, actually three methods.
So method three, no method um method one is going to be just guess if it's close to 50 squared equals 2500.
Therefore, it's going to be the small option.
I have too many things on my screen. I can't see what I'm doing. Okay. If it's close to 60^ 2 = 3,600, therefore it's going to be large. So when I say small, that means it's going to be the 52 option. If it's large, it's going to be the 58 option. So sometimes you can basically just guess. You can say, well, in this example here, 3364 is very far from 2,500, but it's close to 3,600.
So, if I just guess it's going to be the large one, then like um clearly I'm going to be correct.
So, to be clear, this is 3364 like this.
And one thing I've noticed in the uh when I've been coaching groups in mental calculation that students from Asia often don't like methods that have guessing. You want to have a a specific algorithm that tells you the steps for sure. Whereas people in Europe are more happy with methods that involve a bit of guesswork. So I I show you a few different methods and then every person can choose which method is making most sense for them. So method two is compare with 50 * 60= 3,000.
So in general get the 10 multiplied by next tens.
So in this case we know the answer is in the 50s. So we say what is the 10 after 50? It's 60. So I do 50 * 60 or just think of 5 * 6 and that gives me um uh this gives me 3,000.
So to be clear, I'm looking at the 5 * 6 and comparing it to 30. This is the same as doing 50 * 60 and comparing it to 30.
If our square so if um 3364 is larger then it's going to be the large example like 58 and if it's smaller then it's going to be the small version which is 52.
Uh in this example here, clearly now we have the answer with um with method one.
Clearly it's going to be the clearly it's going to be close to this one.
Sorry, clearly it's going to be close to 3,600 and not the other one. And here 3,364 is certainly larger than 30. So it's again we see it's going to be the large one.
also to point out we don't care about the 64 here. I can do this just looking at the 33.
So this is method one and method two.
I'm also going to show you a method three.
And this is the easiest one, but you need to prepare a lot which is just just memorize all the squares up to 99 squares. If you do this then you know the answer. you don't need to do any calculation. You see 3364 and you say yes I know this 58 squared from memory and then you don't need to do a calculation.
So these are the these are the alternatives for finding the last digit when you're doing squares.
If you're if you feel confused by this I recommend method two. If you've know if you know some of the square numbers then maybe learn the rest and do method three and if you feel confident with comparing numbers then you can just do method one.
All of them very easily get you the correct answer.
So let's try uh let's try using method one or method two. If you know all of the square numbers then see if you can figure this out using method one or method two. I'll put an example on the board.
So example is going to be square root of 7744.
So answers in the private chats. What is the square root of 7744?
And if you want to explain your method to the group, then write volunteer afterwards.
Okay, I've got some uh some answers coming through.
Can I invite Kabia, please? [snorts] Kab Kabia.
Hello. Um, so the answer is 88 because um, 77 is close to 64. It's bigger than 64 and less than 81. So I know the first digit, the 10th digit will be 8. And then I take um, four, which is the last digit. So, I know the last digit is either eight or it's two. Um, and I took method one. So, I just guessed because it's closer to 800 than it is to 6,400.
I took the bigger option, which is 88.
>> Good. Yeah. So, when we do step A, we can recognize that not only is this 80 something, but it's 80 and a big number.
So the unit's digit is large and therefore if I got this correct. Yeah, unit digit is 2 or 8.
So method one.
Choose the large option. We're running out of space. Um this digit is 8. So the answer is 88 which is correct. Cool. Very good.
Perfect explanation of how to use method one. Very nice. Let's do another example and I want to show both methods. So for this one try to do method two.
So, for this one, I'd like to know the square root of 676.
Is that what I want? Uh, [snorts] yeah.
676.
Okay. Can we invite please Chatan Dev Pathia? So ch C Ch C Ch C Ch C Ch C Ch C Ch C Ch C Ch C Ch C Chhatan Dev Pathodia.
>> The answer will be 26.
And >> and how do we know?
>> I remember this but I'll state the method also. So it ends with six right and we know 25 square is equal to 625.
So we can conclude that at the end there'll be a 6 and um 6 comes uh 6 comes between 2 square and 3 square that is 4 and 9 and so therefore it can be 26.
>> Good. Yeah. So if here when we remove the last two digits we only have the first six left. So answer is above.
So the answer is between 20 and 30. And can you repeat please? How did you figure out the last digit? We know it must be ending in a four or ending in a six. But how do we know whether it's 24 or 26?
Maybe we need to unmute Jetan again.
>> So I figured it out because 25 squares 625 and since it's more than that it'll be six.
>> Okay, good. So for methods two, what I would suggest for this is to try um actually for this I needum.
So I would do 20 * 30 is 600.
greater than 600 means large option. So it's going to be 26. And what Chatan has done is um slightly different.
It's like method two which is 25^ squar is 625 from memory or maybe you calculate very quickly.
So it must be greater than this. So we have a slight variation here which is also yeah a perfectly good way of doing it. So that again it needs to be 26.
So comparing it to a value that you already know.
Uh okay very good. So this is everything about square roots.
Well most of the yeah most of what's this is the main method for square roots. So it's like cube roots that you get two pieces of information from different digits in the question.
Partly you look at removing the last two digits or for cube roots it was three digits and this tells you what the 10's digit is for the answer. And when you do this for very large calculations you can do this to find whatever is the first digit on the left. For the ones we need for the mental math world cup the answer is two digits. So this will tell us the 10th digit. Then we have a look at the units digit to see what are the possibilities. And for cube roots, we knew immediately what the units digit must be. And we're finished. For square roots, it's actually more difficult because there's usually two possibilities.
And when there's two possibilities, you have to decide is it the big one or is it the small one. There's always one of them less than five and one of them greater than five. And so you can do this by method one just by feeling it's obvious from guessing by the sizes of the numbers or you can do method two by multiplying the 10's digits by the successor of the 10's digit and then comparing those. And if you happen to know the the square of the number in the middle, so 25 squared, you could just use that. That's also possible. And the other thing you can do is if you've just memorized all of the square numbers, then you already know the answer.
So this is the these are the methods that we can use for calculating square roots. I want to show you though a little bonus which is what I use for calculating square roots of very large numbers. You see, for the square root of 3,364, I know just from looking at the 64 that the last two digits are either going to be 08 or 58 or 42 or 92. And I'm going to show you a little algebra bonus which allows me to do that. So, so rule for last two digits of a square number and this is useful for calculating the square roots.
And again, this is a proof using algebra.
So first I going to do an example. So we just did 26^ squar was equal to 676.
Answers in the private chat. Can anyone think of any other numbers? Any other square numbers that end in 76?
Are there any of the square numbers that end in 76?
So, answers in the private chats. Uh, I don't need volunteers for this one. You can just type the type the number.
Okay. So, Vhan has found 74. Um, let's try this. So, 74^2 is equal to 5476.
Yes. Okay. So, this one works.
Which other ones can we find? So um Chhattan has found who already explained one for us before has found also 24 is equal to 576.
So there's three numbers whose square ends in 76. Are there any more? Does anyone know any other ones?
Um 126. Yes, there is actually another one which is a twodigit square.
So, a lot of people finding 24. Very good.
There are some there there is one more.
There is one other one. I'm curious if anybody will find this.
Yeah, very good. So, Vhan got there first. So, this is seven. Well, maybe other people did, but the first one I see was 70 Vhan who found 76. So, well done everyone who find that 5,776.
And then really I only care about the the single digit ones, but we also have um 124 squared, which is I guess I should try and do that. 48 1 5 3 76 um 52 1 5 8 76.
Okay.
So, does anyone notice? Uh actually, let's do another one. So another example.
So 11 squared is equal to 121.
Any other square numbers?
I'll state with two-digit ones. any other twodigit square numbers that ends in 21.
Again, answers in the chat. And then for this question, I don't need volunteers.
How many can we think of?
So again, I still have Vhan's chat window up. Vhan found 39. Very good. So 59^ squ is equal to 1521.
Which other ones [snorts] might we find?
Um Ethan has found 61.
Maybe some other people also told me 61.
Good.
So this is going to be 721.
Are there any others? Can anyone think of any other ones?
Yeah. and Ketchi has found 89 and that's all of them actually these this is everything uh which is 7921 so next question and for this one I will accept a volunteer what is the pattern what do you notice about the four numbers that end the four numbers whose squares end in 76 and the four numbers whose squares end in 21 there's a a special pattern And so at the moment this mathematics is like guesswork. You write you find some examples and you try to see do I notice a pattern? Is there a pattern between these digits?
Uh, okay. I'll ask um Hamdan.
Hamdan is our first volunteer. Our first volunteer for this. What do you notice about the patterns?
>> Sir, uh like there is 50 plus like 24 + 2 uh is equal to 26 and 24 + 50 is equal to 74. 26 + 50 is equal to 76. 74 + 50 is 124. 76 + 12 + 50 is 126. So we can notice this pattern like + 2 + 48 + 2 + 48 and + 2.
>> Okay. Um and so yeah interesting points here. We have the 24 plus uh the 74 is like 24 plus um is 50 + 24 and the 76 is 50 + 26 and then it keeps increasing. So here we have like 100 + 24 uh 100 + 26.
So now we can imagine maybe it's true that the next ones would be 174, 176, 224, 226.
Is that pattern working for these ones?
>> Yes. But here this is different pattern.
11 + 28 is equal to 39. 39 + uh 22 gives 61. And then again 61 + 28 gives 89. And so the next pattern is 89 + 22 is equal to 111. So this will give a number ending with 21.
Nice. Yeah. So, we have a kind of regular pattern that's appearing here.
And so, every every 50 you end up adding making two jumps and then you're 50 ahead.
There's um a pattern even deeper than this. So, I'm going to is um I'm going to see if we have any other um any other volunteers who ask to share. Maybe I can get a couple uh a couple of different ideas for this because there's quite a lot happening. I'm just checking the Yeah. again. Can I repeat? When you send me the questions, don't don't answer multiple times because then I it's difficult for me to know whether um yeah, it's difficult for me to know what you're actually responding to.
Okay. I'm sure there's some volunteers, but I've I'm not finding them quickly.
Um, okay. Um, my three had the same thing, the same thing. You explained to me very well. Nice.
Um uh maybe can we have um Mia if we ask um Mia to volunteer? Do you have maybe we already mentioned what you had to say but do you have um anything else you'd like to add here to Mia?
Sir for uh the pattern I have noticed is that like um the ones digit for uh 76 for the one sending in 76 it's either four or six and for the one in 21 it's either one or 9.
>> Um I'm not sure I understood actually.
Can you explain again please? Sorry.
Or did did we already mute Mia? Um I think she's still talking.
>> I'm trying to unmute her actually. It's I don't know how it Yeah. So for the pattern I have noticed is that for the numbers ending in 76 since their digit since the last digit ends in the six place in six the last digits of the numbers for the square root either end in four or six and then for the one and 21s it either ends with one or nine.
>> Good. Yeah. So you we can see how these arise as part of what we talked about before with by um with the step where we find the units digits. Very good. Cool.
Thank you for sharing. I'm going to show you uh something else [snorts] uh something additional here. So 26 is like 50 minus 24 and this here is like 100 minus 24.
And this here is like 150 minus 24. So we have 24 50 - 24 50 + 24 100 - 24 100 + 24 and so on. And we have the same thing here. So this one here it's 50 + 39 is also 100. Where's the cool symbol?
100 - 11. And this here is like 50 - 11.
And this pattern always works. You always have a number of one option which is less than 25. And then you have 50 minus that, 50 plus that, 100 minus that and 100 plus that and so on carrying on forever.
So let's do final example with this. So if I have for example 12 squares equ= 144.
Can someone tell me all of the numbers all four of the numbers whose squares end in 44? All of the two-digit numbers.
Um, thank you. Yeah, there's a Thank you for correcting me. Yes.
Okay. And if you found all of them, then tell me you volunteer.
Okay.
Uh, okay. So, we have a correct answer from Sanina. Sanina from Sri Lanka.
You found all of the other ones after 12.
>> Yes. And there.
>> Good afternoon, sir.
Good afternoon.
So the other twodigit numbers where there are squares ending double 4 are 38 squared >> and because I found out that 50 - 12 is 38 >> good yeah >> 62 squared >> well 50 + 12 is 62.
>> Very nice. Yeah.
>> And then 88 squared based on the fact that 100 - 12 is 88.
>> Very good. Yeah. Perfect explanation.
Thank you very much, Senator.
>> Thank you, sir.
So this means that if we have uh let's say if I'm finding in the red example the square<unk> of 7744. Another thing I can do is say well it ends in 44. It's like 144 which is 12. And so it's going to be one of these examples. It's going to be 12 38 62 and 88. And clearly it's this one.
This is also really helpful if you want to learn if you want to memorize the square numbers because when you learn 88 squared you know that the last digit must be 44 cuz it's like 12 squared. So you get this [snorts] removes the work of memorizing by about half. You have about half as much work to do.
So why is this rule working? So um so example with um let's do with uh 12 144. So it's any um multiple of 50 plus or minus plus or minus 12.
So we're looking at squares of any multiple of 50 plus or minus 12. So this would be 50 n. This is any multiple of 50 + or minus 12. And I'm going to find the square of this number. So this is when I'm using algebra by using foil or expanding brackets or binomial theorem or whichever method you use for expanding the brackets. This would be 2500 n^ 2 plus or minus 50 * 12 doubled.
So double of this number and then either + 12 squared or -12 all squared. The double negative makes a positive. So this will always be 144.
I'm going to do it as 12 squ. This then is equal to I'm going to copy the the same thing. This is 144 and 2 * 50 is 100.
Now I can do the factoring and this is 100 lots of 25 n^2 plus or minus 12 + 144 equ= a multiple of 100 + 44 therefore ends in 44. So this here proves that any number um that um if I have 50 plus or minus sorry I start again this proves that if I have a multiple of 50 plus or minus 12 which are the examples that Santha just told us then the square of that number must end in 44.
And we can apply the same logic not just for um 12 but for any other number. I'm going to copy paste. I'm going to change the color. So example if I have um any multiple 50 plus or minus x where um let's do plus y so y can be any number then I just replace this this is going to be um y here and y^ 2 like this. This is going to be some multiple of y. It's a different multiple of y actually doesn't matter. Plus y^ 2 which is a multiple of 100 + y^ 2. So, same last two digits as y^ 2.
And because my text is all running into each other, I'm going to delete that.
So, this proves it now for all numbers for all of these patterns.
So, okay, what have we done here with square with um square roots? So first of all we saw that we could do a method like the cube roots but for square numbers but the units digit we had two possibilities. So we needed to have some methods and we have a couple different options here for finding what the last digit would be. Then we have method three which is just memorizing all of the square numbers. And so to help us understand the structure of the square numbers, we have a look at the last two digits. And we can use algebra to prove that the rule is true. That the numbers with the same last two digits always fit this pattern. Multiples of 50 plus or minus whatever amounts. The [snorts] example with 12 is you've got 12. 50 - 12 is 38. 50 + 12 is 62. and 100 - 12 is 88. So if you know if you're looking for a square number that ends in 44, if you find one of these numbers, then you can easily find the others.
That brings us to the end of what we're doing with the um cube roots and the square roots. [snorts] We're um I think um Ainach, you wanted to have a short break.
>> A short break. That's right. Um, I think this is an opportunity >> until 5:00 p.m.
>> Yes, let's do that.
>> Seven minutes.
>> Let's do that.
Let's have a short break and and circle back again.
>> Okay. Yep. So everyone take a a little breather for a moment and then be please be on time because we will already be starting at [snorts] um if you're in India this will be 5:00 p.m. and if you're in other countries it will be 30 the hour. So at the hour then we start again which is in about 7 minutes time.
Okay, welcome back everyone. We're going to start in the next 30 seconds. Uh to begin, I've put a little question on the screen. I have a really really really really big multiplication and I want to know what would be the last digit of this multiplication.
Okay. Uh, let's have Tobeti or Ragwood.
Tobeti, please.
Lots of correct answers coming in. Okay, I think everyone finds this question quite easy. Uh, Tobeti, what was your answer and why do we know it's true?
So, so my answer was four because um if you take the last digits of these two numbers which are three and this and when you multiply them by each other since um since the the other digits after they don't they don't really matter in this in this case. So um 8 * p will be 24 and so that means that the last digits of this of this um equation is going to be four.
>> Very good. Yeah, it's exactly like this.
You just need to look at the last two digits and just very quick algebra if anyone wants. We have some multiple of 10 + 3 multiplied by some multiple a different multiple maybe of 10 + 8. And if I multiply the brackets, I end up with 100. N + um 30 m + 80 n + 24, which is equal to basically the first three numbers is multiples of 10. So anything that isn't just the three and the eight is a multiple of 10. So n m + 3 m + 8 m + 2 and then with the four well let's just remove that plus 24 so therefore must end in four.
So this is a trick that can help you with multiplications that even without doing the full multiplication you can look at the last digits which uh the last digit of the two numbers you're multiplying and that tells you the last digit of the answer. So what we learn here?
Look at the left digits of the numbers.
Do you multiply?
That's the Yeah, that's the last digit of the answer to the multiplication.
So, a very simple trick and it's working just the same as for the squares and for the cube roots.
Okay, nice. So, how do we multiply numbers together? So, you will have to do multiplications for the mental math world cup.
And the method I would recommend is called cross multiplication or sometimes known as foil.
So example and I did this yesterday in the master workshop. So today I will do it um quickly. I won't go into much detail because we did a lot yesterday.
And I will give you a link to a resource on my website that has more information about this so you can look at it yourself if you would like. So example, let's do 54 ultiplied by 67. It's the same example that I did yesterday.
And I will explain. So if you want to answer, you can, but I'm going to explain the methods here. So it's 54 ultiplied by 67. And I'm going to give lots of space here. So the first thing I'm going to do is I'm going to multiply together the four and the seven. And this tells me that the answer must end in an eight for the same thing that we just heard.
So we have units 4 * 7 = 28. So this means I can write the digit 8 in the answer. So write eight. What's happening with the 20? The 20 is how many tens I've got. The two I've got two tens. So I'm going to carry that to the next stage.
So at the moment we know that the answer is going to end in eight.
Hence the jets. I'm going to start with the carry. I always start with the carry.
I'm going to write this.
Uh yeah. So I always start with the carry from the previous one. And then I'm going to expand this blocks to cover everything. And I'm going to have a look at this cross in the middle.
So the five * the 7 and the 6 * the four.
So I um I do 5 * the 7. And so I don't get confused with the numbers, I already think of the answer. So I start with two. I add the 5 * the 7 and I've got 37 in my mind.
Then I do the other part of the cross which is 6 * 4 and when I add that on I get 61.
So this means I have 61 10. [snorts] So I can write down the one and I can carry the 6 10 into their units. So the answer is now I already have the eight. Now I can put the one there as well.
Now I have the hundreds. So I have the six that I carried. And I now move my box fully to the left.
And I How many hundreds? I've got five.
Oops. 5 * 6 hundreds. um that there is equal to 36 and now I can just write down the rest of the answer.
So this method is called cross multiplication. You may have seen this in algebra class related to foil.
There's a few different it's also one of the the vdic sutras explains this as well. And this is how I recommend doing two digits times two digits.
we um we start off looking at these ones. So actually this is maybe easier to describe if I do it in different colors. So the pink is finding the units here.
Then I'm going to have my cross in the middle. Let's do the cross in orange.
And that will find me the tens.
And then finally the hundreds. Let's [snorts] do this in yellow. So I look at this afterwards.
This works then for all uh all twodigit times twodigit multiplications.
When you see the questions, they're likely to be well, no, they are going to be written in a single line. So 54 multiplied by 67. Let's add lots of spaces to this. Okay. So 54 * 67. How does the same method look in this example?
So the 5 * the 6 is [snorts] like this.
The 4 * the 7 is like this. And then the cross is like this. And it's a bit like a face. It's a bit like a smiley face.
We've got the the nose um the nose between four and six, the mouth between five and seven, and then two eyebrows.
One of them [snorts] is pink and one of them is yellow.
So when I'm doing this, I first of all do the pink four * 7 to get the units.
Carry the 10 left over. Then I look at the bottom. I do the middle ones 4 * 6 and the outside ones 5 * 7, whichever order I want. add those with the carry and that gives me the 61 and then to do the hundreds I go [snorts] to the left side of both numbers. So I'm looking at the five and the six and this then tells me the answer. So this is how cross multiplication works and we have here three diagrams of how to think about it.
I'll also give you um a page on my website so you can read more about this in a moment. Let's just do an example together.
So the example I want to do is what is 43 multiplied by let's do 56. So 43 * 56 answers in the private chat and you can say volunteer if you'd like to explain how you use this method for solving it.
A question came through. Yeah, very good. What hap what if we have to do this method with three-digit multiplication or more? You won't have to do threedigit times anything greater than one digit.
I'll say something about three-digit time one digit in a moment. Does this method work for like three digits times three digits? Yes, it does, but it's um more steps but quite simple once you learn it. I'll give you a link on my website that shows you how you can use the same method for doing multiplications of three digits, five digits, eight digits, any number basically.
>> Hey Daniel, uh for for Middlemath World Cup, we will have uh three three digits by three digits. We also have u you know >> Okay.
>> Yeah. Yeah, we do have three digits by three digits.
>> Ah okay. Thank you for um for informing us. Okay. So I'll explain >> for grandmaster and Zen master.
>> Okay. Yeah. uh the examples I saw didn't have. So I I clearly didn't see enough of them. Okay. So I'll show you in a moment how we do this for larger ones like three-digit times three digits.
So a lot of answers coming in that are correct. So I'm looking for someone who has the correct answer and who would like to volunteer.
Okay. and we um try and pick as many different people as possible.
Um So, we're not um Okay. Can we have a Mia, please?
I got 148 because I multiplied five then like so can I write can I explain like can I write on the board >> easier just to explain rather than writing so what did you start with >> I feel more comfortable writing while explaining that's why >> um >> okay I think >> I think best not writing best not writing But it's okay. Thank you.
>> Um >> yeah, for the I think it is possible to write on the board, but for speed we won't um uh we won't um we won't be writing on the board.
Um do you want to explain or should we invite somebody else?
How many >> sir? I got 2,48 since first I multiplied four with like 56 which equals to 224 >> and then I multiplied three with 56 which will give 168. And now since we can't just like put them together like 2 4 168 since it'll become bigger than needed. I I took the last two digits of 224 which is 24 and I added it with 16 from 168 and that's how I got 2,48.
Okay, so this is uh this is actually using a different method but it's still valid. So um and this only works for small numbers but we can totally do it.
So here we did um 40 * 56. We also did 3 * 56 to get 168.
And then we add these together. And the nice thing is that the first number always ends in zero. So adding these together is really this feels like doing um 224 + 160. And this gives the this gives us the correct answer. then 2,48.
So this is an alternative method. It's not the cross multiplication but some of you may also like to do this methods.
It's um it's completely true. Um Retif can we invite Ratif to also share a solution to this.
>> Morning sir.
>> Morning.
How did you solve this one?
>> So, what I did is I got my pen paper and I thought 3 * 6 was 18.
>> Good. Yeah.
>> So, I wrote the eight and carried on the one.
>> Then um then I crossed it. So then it was 3 * 5 and 4 * 6 which was 15 + 24 and then we add the one that we carried on which gives me 40.
>> Good. Yeah. And my recommendation for everyone is to always start with the carry. So at this point I think maybe you do differently but my recommendation is to add start with the carry and then at this point I already have 16 and then at the end I will get the same answer as you. Uh what did you get?
>> I got 40.
>> Good. Yeah. So we have 40 tens. So I'm going to write what I write.
>> So you write the zero and carry on the four.
>> Perfect. Yeah. Very nice.
Then I did the last >> then the last line was the four times the five.
>> Good. Yeah.
>> And then I just added the four which gives me 24.
>> Good. Yeah.
>> And then what's 24?
>> Very nice. Yeah. So we got the eight. We get the zero eight. and now it's 2408.
Okay, very nice. Cool. Thank you for sharing. So, this is how we use cross multiplication. And we've also seen an alternative method that some people might prefer which works for smaller numbers. Um whereas cross multiplication works even if you do like 20 digits times 20 digits.
So, okay, how can we extend this then for very large numbers?
Uh so you have to go up three digits times three digits. Let's do let's just do three digits times three digits. So imagine I'm doing 234 multiplied by 567.
So I write everything very big.
So uh Here is a lot of steps to write out. So I'm going to do it more compact.
So I start again with the right and I'll just get my space. Yeah. Right. So I start with the the units digits.
Why is this not the color that I wanted?
Here we go. So we start with the units digits which is like before 4 * 7 = 28.
So, write eight. That's not how I spell.
Uh, write eight and carry the two.
So, so far everything is the same. Now, I'm going to get my digit that I carried. Oh, yeah, maybe I can write the answer.
So, it's eight. So, here I can get the 10. And then I can expand this in this direction.
And then I'm going to do the cross, but I'm going to illustrate the cross in a different way.
So I've extended this to to the six. So I'm going to think of the opposite side of the cross. And then I'm going to imagine that I'm moving this this star in opposite directions.
So it starts off with the 6 * the four like we had before.
And in my mind now I have 26 because I already add the two. Then I imagine that the star moves one place on the bottom it moves to the right and at the top it moves to the left. And then at this point I'm going to have my 3 * 7 which is going to be 47.
Just like before, I'm going to write that seven and I'm going to carry the four. So, this is going to be something something 78.
So, so far everything is the same, but the hundreds is now going to get more complex.
So, I move the this box.
So, I'm looking further. The star moves along the bottom. But now these arrows they have to move further. These arrows have to move in opposite directions across the whole box. So [snorts] I start with my carry and now I have to do the five * the four which gives me 24 plus. And now imagine that the stars move to um I'll change the symbol. So they move to this position.
So that's going to give me 6 * 3. Again, I'm going to get confused with all of the numbers. So I'm already going to add this on. So 24 + 6 * 3 18. 24 + 18 is 42.
And then finally, I move the stars all the way to the other side and [snorts] I end up with uh let's put a heart here.
So I end up with these ones.
So 7 * 2 and that gives me 56.
And now finally like before I can write the six and carry the five and the answer is something something 6 7 okay 6 7 8 in this case.
So, it takes some practice, but you're always um you're always looking at the um you're always doing it in this balanced way, finding the next two numbers that you need um that you need to consider and then moving in opposite directions to find all of the others.
At this point, of course, we haven't finished. So the thousands we have the the five that we carry and then the box is going to shrink. I'm actually going to change the color of the box. So I'm going to do it like this [snorts] so I can remove it later. So I'm now looking in this box. The opposite corners are now five and three, which gives me 20. And if I move to the other corners of the box, then I end up with the 6 * the 2, which is 32.
Again, write the two, carry the three.
And this should give me something. 2 6 7 8. And then we just keep on going like this. Now we're almost finished.
[clears throat] We've got the 3 plus the 5 * 2 which gives me 13.
So the answer is therefore 132 678.
I just check on my calculator to make sure I didn't make a mistake somewhere.
And yes, this is correct.
I'm going to illustrate with the symbols where these came from. So this was that bracket.
This was that bracket and this was that bracket. So the diagram on the left is showing me is showing you the 100 step, but you're repeating the same idea in your mind each time.
How does this look like with the digits written out?
So you will have your 234 multiplied by 567.
So now I can do a diagram like we saw on the previous screen for 54 * 67. How is this going to look?
I'll use um different colors. So we start off with the I do it with the stars.
Okay. So we we start off with the stars above the four and the seven and then we multiply them. Then on one of the numbers, we move the star across.
And imagine that we're moving these now in opposite directions. So this moves from the three to the four. This moves from the seven in this direction. But we only uh when we get to the six, we can't go any further because 234 doesn't have any digit after the four to match with the five.
Then for the next one, we move the star to the two. And again, we're going to move these in opposite directions to get the um the the 4 * 5, 3 * 6, and the 2 * the 7. So you can imagine the same thing for these numbers written out in a line.
So I recommend to practice using this methods to um to multiply uh to multiply the numbers and with practice you can get very fast. So I'm as a demonstration I'm going to show you uh oh actually I don't ah no okay I I can't do the demonstration because I didn't I couldn't get the I have a new laptop and I don't have the correct multiplication software installed. But when I practice this, I can do three-digit time three digit and it takes me about I think it's about 12 seconds and then fourdigit time fourdigit I can do in about 22 seconds.
And the fastest people in the world, they can do 8digit* 8digit and they get the answer in less than 1 minute using this method. This is more advanced than you need to know for the competitions but you can see you can use this for solving any kind of multiplications and with practice very fast.
>> Yeah. And just for context for everyone you know some of you may not have are looking at this for the first time and probably are thinking that you know this is take this is too long or you know it's not the way I I'm used to doing in school. But if you if you if your intent is to be a mental athlete, all right, this is the way to do it so that you can do it mentally and not uh you know write it down and do it. So, so uh these methods that Daniel is showing is the psychology of how a mental math thinks and how he solves the numbers mentally.
>> Good. Yeah. Thank you, Aan.
So this is maybe what the more complex method that we've had a look at today and I don't want to do like too many examples of this because it becomes not interesting for the workshop.
>> Um so what I have is a link on my website which explains this method with I think it's fourdigit time five digit. It's working the same way. It doesn't matter how many digits are in each number presented in a different similar way with different colors. And then you can try doing this yourself with a pen and paper to see if you can get the right answer.
Uh what else is there to say about multiplication? Um not much.
Uh >> I think we can move to the next topics Daniel if if because you know we can have these uh these on recordings to for everyone so they can come back and review this you know you know at at their own convenience and we let's cover the other topics then >> also useful yeah okay so the the two main topics to cover now are division and then highest common factor and lowest common multiple the last two they come together so I'm I'm going to do division next And my um a big theme of this workshop we had with the cube roots and the square roots and the multiplication is that you can get some information from the last digits.
So, um, imagine I have a number that ends in six and I divide it by a number that ends in seven.
What would be the last digit of the answer?
Answers in the private chats. Also, I've just told you the answer. Let's do a different example. Uh, I have one that ends in four. I divide it by a number that ends in seven.
What would be the last digits of the answer?
So, what's the missing digit? And I take a volunteer for this question.
Okay, good. I've got some some correct answers coming in, which is very nice.
So, I take a correct answer from someone who likes to volunteer.
Okay. Write the word volunteer if you'd like to.
Yeah, we have number 67 lots of times today. It's a a useful number.
I'm still waiting for someone with the correct answer to this question who would like to volunteer.
Okay. And can we have Ali Ahmed, please?
Ali Ahmed.
>> Yes, sir.
>> Yeah. What do you think the last answer, last digit is? I think it's two because as you said for the second digit last number is seven.
So I can think that in the time table of seven it's 7 * 14.
So I think the answer could be two.
>> Good. Yeah, you're exactly um correct.
So it is it is two. And we can look at the seven times table. 14 21 28 35 I'll write all of them out. 49 56 63 and 70.
So here they all end in different last digits.
And the only one that ends in a four is 14. And so this tells us that this must be the correct answer.
So yeah, very good. Uh because we know then if we multiply a number that ends in um two by a number that ends in seven, the answer would be a number that ends in four. And the only way we can multiply a number by something ending in seven to get something ending in four is two because all of these last digits are different. Let's try another example like this. So, I've got something that ends in, let's say, um, two, and I'm going to divide it by a number that ends in eight. And I want to know what is the last digit going to be.
I give you a clue. It's a trick question. This is a trick question.
Okay. Had the first correct answer. I'm waiting for a correct answer with a a correct answer with someone who also wants to volunteer.
Most people writing the same thing.
I've only actually seen one correct answer.
Lots of answers coming through. Only one of them is fully correct. Two of them.
Now we have a second answer fully correct.
I've responded to both of those that I've seen.
Okay. All right. Can we have um Alex, please? Alex sh.
Uh yes. Hello. [laughter] So the answer could be ending either in four or nine because looking at the time table, yeah, it's either 8 * 4, so that would be 32. Or it would be 8 * 9, which would be 72. And since both of them end in two, they're both um possible answers.
>> Perfect. Yeah. So we like uh this we look at the eight times table looking for things that end in two and it could be 32 or it could be 72. So if those of you who gave the answer four it could be correct. Those of you who gave the answer nine it could be correct but we don't know which one. It could be either.
So it could be um ending in um four or it could be ending in nine.
So specifically this could be like 32 / 8 is 4 or it could be like um need to move this like 72 / 8 is equal to 9. We don't know which one.
So this is important for division because when we're doing the division, we can look at the last uh we can look at the other digits which I don't show here to have an idea approximately what the answer is going to be and then we can get information from the right of the question to figure out which one it would actually be.
Um, so, uh, as an example of this, I'm going to do one that's fairly, um, fairly tricky.
Yeah.
So question what would be this number divided by divided by 38. Eight.
Again. I can take I can take volunteers for for this So, what's the full answer? We got a couple couple coming in. Uh, can I ask um sanin sanin please?
So find out 3,268 divided by 38. I started from the left. So first I found out on how many 38s were in three. None. So I left that blank.
Then I found out how many 13ths were in the number 32. Also nil. So I decided to leave that blank to find how much 30th were in 326.
Then I found out that that 308 * 8 is 304.
So then I write eight at the top. So that's going to be the 10th's place.
And then I minus 304 from 326 to get the remainder of 22.
Then >> I brought the the eight at the end of of the dividend 3,268 to the 22 forming a new number 228.
Then I find out how much 13th were in 228.
And then I found out that 13 * 6 is exactly 228. And when I minus 228 from 228, I get I get a So therefore the answer to this division is 86 with no remain.
>> Very good. Yeah, thank you very much for explaining. And the answer is 86. So working from the left then you use in this case multiples of 38 to find what's the first digit going to be. And so we can maybe estimate it's going to be um like 90 or 80 or something like this. So you find one of these and this tells you what the 10th digit is going to be. This takes a bit of work, a bit of thinking but um but we find here the answer is going to be 80. And we also find that the remainder is 228.
So the remainder is quite large.
So the the um compared to um yeah so I'll come back to that then working from the right. So um Sanin did this pretty much just by guessing the answer and uh and checking it. Here we can say that the last digit must be 1 or 6 by looking at the 8 * table and seeing what ends in 8. 8 * 1 is 8. 8 * 6 is 48. So it's one of these.
So we just need to decide which of these it's going to be. And because the remainder in this case is 228.
We we don't even have to do the division to check. 228 is not 38 * 1.
It's much bigger than that. So it must be multiplied by six. And we can just guess then that the answer is six. So it's like with the square roots we had two possibilities and we just decide is it the big one or the small one. Here we're doing the same. Is the remainder something big or is it something small?
And then you choose the option.
The other thing to note here is that for for square roots we had um yeah for for the square roots for method two the options are symmetric. So if you have one, you have nine. If you have two, you have eight. If you have four, you have six. It's always symmetric around five. Here it's different that the options are always five away from each other. So if one option is one, the next one is six. If one option is two, the next option is seven. And this is nice because they're always far from each other. And so the the method for uh when you do the multiply multiples of 38 for the division this means that you can be lazy with the last step because it should be obvious.
It should be quite obvious whether it's the smallest digit or the last digit.
Um so this is the main tip and trick I want to show you for the divisions which is by looking at the last digits you will have um what am I saying? Yeah by looking at the the last digits you will have either one or two options usually of what the last digit is going to be.
So if you're dividing by a number that ends in one, a number that ends in three, a number that ends in seven, or a number that ends in nine, there's only one option, which is very nice.
If you're dividing by 2, 4, 6, or 8, then there are two options. Well, not really options.
Two possibilities and they are five apart.
What happens if dividing by five? This is the tricky one. So, if it's one, if you're dividing by five, then you end up with actually five options, which is very tricky. And this means you have to be more precise with the division algorithm because the left hand side of the question is going to make it harder. So when you're dividing, the easiest ones are when the number you're dividing by ends in 1 37 or 9.
And 2 4 6 and 8 is more difficult. And dividing by five, this is most tricky.
So this is my main tip and trick for the division methods.
And >> hey Daniel before you move ahead would you be would you have anything to offer for for remainders also for them? Is is there is there any tip or trick for remainders?
>> So the remainder is for the um it's just a subtraction at the end.
>> Yeah. So in this case here we if we think it's going to be 80 something you say what's 38 * 80 and um yeah and then compare it to that there's there are some shortcuts I do but there are a bit I think there will be more confusing to explain. All right.
>> So, um, but that does bring me very smoothly into the last thing I wanted to mention about division. And this is something that a lot of mathematicians get confused by or they don't know, but you will need to know this for the mental math world cup because the exam uses the proper terms for the for the the division. So, if I have um I'll just pick a simple example. Let's say 100 / 7 = 14 remainder 2. Each of these terms has a special name.
Could anyone tell me what's the special name for the answer without the remainder?
Does anyone know this?
begins with a Q.
Couple people saying it. Um, yeah, Sanin, that's um, it's just a one word answer. I'll let you you tell us.
Yes, Senator, you're unmuted.
>> Senator, you are unmuted. You can you can see the answer.
>> Yeah, Senator, we can't hear you actually. I don't know why.
Do you want to pick someone else?
>> Um, it's okay. I'll I'll pick my I'll pick myself for this one. It's a one word answer. So, it's the um quotient is the answer.
So, the quotient is the biggest number that it divides into and the part that is not fully divided. This here would be the remainder.
What's the term in pink called? It's the things that we are dividing by.
So answer in um answer in capital letters and say volunteer. So answer in capital letters for this one and say volunteer. So I know you're answering for the correct one.
Okay. Yeah. Oh yeah. Now I see a lot of people had the right answer of quotients for that one.
So answer in capital letters and say volunteer if you think you know what this [snorts] one is.
Um okay. Can we have um cartic cartic siy >> divisor?
>> Good. Yeah. So, this would be the divisor.
So, it's the number that you're dividing by. And you will see these words used if you're reading anything about uh about uh mental math divisions. And also some of these words are used in the competition.
And I'll just fill this one in myself.
This one is the dividend.
So the dividend is divided by the divisor and that gives you the quotient.
And there might be a remainder or the remainder could be zero.
So it's important to know these words.
So I just remind you them here.
Okay, that's everything I want to share here with the division method. The thing we kind of skipped over a little bit is the method working from the left. So I will um at the end of the workshop I have a few little little gifts for you which are pages from my website that you can use for uh for studying some of these things. if we went over them too fast today. And one of them is a method for the for the division algorithm that actually allows you to divide by very large numbers. So that's a more complex method than you need for this. But if you want to learn something more advanced, then you also have that.
>> I think Daniel I think Daniel you've used that method in a in one of our previous sessions. So we can share the link for that. Yes, we can share the link for that webinar also with the with the participant. We can have a look at that in that from there.
>> Yeah, there was one time we did like the full full thing with the flag methods.
>> Yes, >> it's um it takes quite a lot of time to to go through but yes um yeah, also a good idea. Yeah, let's finish then with highest common factor and lowest common multiple. And I have two very clear tricks for this. [sighs] So first of all, what does highest common factor mean?
It means the highest number that is a factor of both numbers. So for example, if I have um let's say 150 and 90 common factors of both of these numbers, both of these are multiples of 10. So I can say actually 10 is a common factor.
Two is a common factor. So lots of these numbers are common factors. Um actually even 15 common factor um you can find some more but the highest common factor is 30. So it's the largest number that divides into both of them, both 90 and 150. And when you learn about these at school, they say find the full prime factorization. So 150 is going to be 2 * 3 * 5 * 5. 90 is 2 * 3 * 3 * 5. and then count the number of different prime factors. But this is actually a bit confusing for um it is not a very efficient method for mental maths. So I'm going to show you a special method which is based on Uklid's algorithm that simplifies it. So this is this is my method.
So the quick trick method for HCF.
Subtract the numbers from each other or any multiple of one number from the other number.
This gives you a smaller number.
So example I could do 150 take away 90 which gives you 60.
You can repeat that more times if you want. Usually in these questions it is not necessary. And then finally find the HCF of the US number with one of the original numbers. So example, we want the HCF of let's say 90 cuz it's smaller and 60. And now it might be obvious that the answer is 30.
So let's do an example using this trick and I will think of an example which is where this trick is very helpful.
So, HCF of let's do uh 112 and 114.
So using the quick trick method, how would we find the HCF of those? And I also look for a volunteer.
Okay. Lots of incorrect answers actually. Interesting.
Okay, I have seen one correct answer.
Lots of incorrect answers.
Of course, I don't see everyone's answers at the same time, so you might have answered correctly.
So the answer is not 12 and the answer.
Okay. Okay. Now we have one. So Matthew Schultz. Can we invite Matthew Schultz?
Okay. Now I'm seeing more correct answers.
Matthew Schultz.
>> Am I?
>> Yeah. How did we How did we solve this?
I took 114 then I minus it with 102.
>> Okay.
>> Then I go 12 and the biggest FC FCF is six.
>> Good. Yeah. So the HCF must be >> uh it must be it must divide into 12. So 12 must be the HCF must be of 12. So is it going to be 12? Uh no no because 112 is not a multiple of 12. But it could be six >> is.
>> So I can do HCF for example of 102 and 12 and that's easier to find that the answer is six.
Okay. Very good. Thank you for sharing.
um optional um I can also do an optional repetition. So here I can do 102 subtracting um 96. Why 96? Because this is a multiple of 12 and this gives me six. So the HCF must be a multiple of six. And actually, yes, it's six because 102 / 6 is uh a whole number.
And this is much easier than trying to find the prime factors of 102 of 114 and 102.
I'll do one more example and then I'll tell you a thing for the lowest common multiple and then I will give you the the bonuses. Then we finish the theory and I give you the bonuses because that will then be the end of the workshop. So the next example is the highest common factor of let's do 304 and how should I do this one 304 and [snorts] actually I've changed my mind. Okay, I'll do a,01 and 98.
So what would it be? A, 101 and 98.
I'll give you a clue as well. We can find the difference between these numbers, but it's more useful to find the difference between a multiple of 98 and 101.
Okay, this time the first person I checked had the right answer. Put the word volunteer afterwards if you want to explain.
Okay. Uh can we have rayanch rayanch to explain please?
So the way I found this answer first was like I subtracted 98 from 10, 101 which I got the answer as 93.
Uh and one thing I noticed like I didn't do the exactly the same method as this one >> but I noticed one thing in both that 98 was divisible by 49. So then I tried it out for 93 but it was not and then I reduced from 49 to 7 and then I tried it on 93. When I turned seven on 9003 it was divisible. So I got the HCF as seven in this way.
>> Okay. So in this point it was almost like the question was too easy and without even doing a big multiple um you you found the answer. Very good. And so if you know the factors of different numbers then it can help a lot. In this case here the fastest shortcut is to take away 980 and this gives me 21. So the highest common factor then I can just do with 98 and 21. And what do I know about these numbers? They're both multiples of 7. One of them is 7 * 3.
One of them is 7 * 14. Doesn't share any factors. So therefore the answer is going to be just seven. And this means I don't need to prime factoriize 1001 which is uh maybe you know the answer but it's um it's more work to do that.
So this is my big trick with the highest common factor.
Let's finish then with my trick for the lowest common multiple.
So if I have the highest common factor of a and b where a and b are any numbers and I multiply this by the lowest common multiple of a and b the same numbers this here is always equal to a * b. This is always true for any numbers a and b.
Therefore, if I can calculate the HCF of the numbers, then the lowest common multiple of A and B is equal to A multiplied by B divided by the highest common factor of A and B, which maybe seems difficult to calculate because I have to multiply A with B. But I don't actually because here what I can actually do is I can divide A by the highest common factor of A and B and then just multiply that by B.
So this is kind of in algebra. So as a method I can say find the highest common factor divide one number by the HCF but multiply by the other number. So let's try this. So what would be the highest common factor of 101 and 98?
Well, that's not what I meant. I meant the lowest common multiple. How do we find the lowest common multiple of these numbers? So, this here is going to be um first step is find the HCF of,01 and 98 which is 7. We did that before.
And then the LCM of these numbers is then going to be 98 / 7 * 1,01 which is just 14 * 1,01 which is equal to 14,014.
Maybe a bit of a crazy example because it was from the the last one we did before. But yeah, once I know the HCF, I [snorts] can make the 98 smaller by dividing by the HCF and then multiply that by the other number 1,01. And this gives me the answer. This is so much faster than trying to find the full prime factorization of both numbers, collecting the prime numbers, and multiplying together, which is what you would be taught in a textbook at school.
This is a a much nicer shortcut. So let's finish with an example of this.
So use the shortcut to find the LCM of let's do um 125.
Um, make it easier.
126 and 100 and 54.
This is a a pretty difficult one.
Yeah, this is a pretty tricky one, but if you follow this method, it's actually not that hard.
And again, I accept a I accept a volunteer for this one.
Okay, we got some correct answers coming in. Also say volunteer if you'd like to volunteer.
And then the first time I see a correct answer with a volunteer, then we'll go with that.
Um, okay. Let's invite Alex back. Alex Sh.
>> All right. So um if we follow again the methods so what we would get is um so first we do 154 and then do minus 126.
>> Nice. Yeah.
>> Yeah. And that will give us 28.
However, if we were to check so for example if you do 154 / 28 uh we get 5.5 if you like do the full calculation. So obviously it's not correct. The thing is since it's 0.5 we know that if we divide it by two um it'll give us a whole number. So we will do uh 154 divided by 14 which um one sec gives us 11. So after that we multiply 11 by 126.
So 11 by 126 and that will give us 1,386 which is the lowest common multiple.
>> Very nice. Yeah, perfect solution. Very nice. So we first of all find the HCF using any method we want and we now have the trick for the HCF. If the numbers are close to each other, we can just subtract them and then um maybe we have to check maybe it's not the first thing we find but the second thing we find.
The next thing we do is once we got the HCF and we checked it by dividing, then we just multiply that by the other number and that's already the LCM because of this special formula that the LCM is one of the numbers divided by the HCF multiplied by the other number. And in most cases, this is a very nice trick for solving the LCM.
Okay, that's the end of all of the theory for today. Um so yeah thank you for joining today and going through all of these these methods looking at the cube roots the square roots the multiplication we see the importance of addition and subtraction through all of this. So um everything you do practicing addition and subtraction is of course important and division and the highest common factor and the lowest common multiple which covers all of the more difficult um categories that we have in [snorts] um in the mental math world cup.
Um, I'm going to drop some links in the public chats which are the kind of gifts from this. I'll take me a few moments to do that. Meanwhile, anything else ain we want to say just to finish for today?
>> Sure. uh you know first of all Daniel thank you for this wonderful session.
I'm sure all of us over here have have really found some amazing ways to look at these calculations in a different way from what we've learned uh in schools.
Uh so so participants u as as Daniel mentioned he'll be sharing a few resources right now. We will make sure that the same is shared with all of you tomorrow along with the recording of this session tomorrow. Uh and uh also with regards to your your upcoming mock tests uh that are going to that is going to be happening between the 24th to the 27th that is a practice test. Uh again that's not uh the scores over there are not going to be counted. So don't worry about u you know uh uh how that will impact your your u your qualification later. So the round so the mock test is between the 24th to the 27th and the round one of the standard mandatory challenge is between the 1st to the 5th of August. So please prepare for that.
Obviously those who have made it to the top 200 of the mental math world cup in 2025 or have been the top 100 in the GMMO Dubai or GMO Mumbai or Bahrain Mental Math Olympiad in Mumbai in in Bahrain will make it directly to the round two the grand finale so you will not have to go through the round one those who have made it to the top 100 in the standard mandatory challenge. So uh I'm sure you must have received a communication about the same also. Uh u just in case you have any doubts just drop us a message on plus918291027238 or drop us an email on contactivemcl.com and we'll be happy to answer any of your doubts uh regarding your participation.
And just to finish then with information about the the bonus links. So I've put links to five pages from my website that are relevant to what we were doing today. And in fact maybe I also do the um there's a a sixth one I'm going to add if I can find it. Yes, this one here.
So, six uh [snorts] six little bonuses from my website. I'm just going to share my screen and show you me using uh a couple of these. So, first of all, I have a way of practicing the the small multiplications that we did today.
So, screens basic multiplication trainer, you have the link to this in the chat.
So if you want to practice twodigit times two-digit multiplications or even smaller ones the this is how we use it you press enter you try and solve these [snorts] in whichever way um you're going to do um so in this case if I can do maybe the 8 carry the one 22 46 [snorts] 4 carry that 324 68 or maybe um in this case maybe there's a different trick that I want to So I'm going to do 258 from that 4042.
So there are some other tricks you might know for multiplying these. Every time I'm just pressing enter. I have the question here and I type in the answer as fast as I can. And it tells me what my average time is and what I did for each of the questions. And if I got any of them incorrect, let's get one of them incorrect. So I type in this one. it goes orange and it tells me what I got wrong. So this is how you this is one extra tool as well as all of the apps that um live MCL has provided you can use for practicing their multiplications.
And the other one is if I just find I have it somewhere of the flashcard store. Yeah, here we go.
Somehow I sent you the link but didn't have it open on my computer. So this is a flashcards tool and the way you use this is slightly different. So you pick something you want to practice. So let's do the cubes. So I type the answer and press enter. I type the answer. It gives me the same one again. Press enter. 64 3375 512 and so on. If I get one wrong, actually I've forgotten that one. 2197.
Then it goes orange. There's actually a timer for 30 seconds happening in the background. It's not happening now because I'm between the questions, but it's happening when I'm answering the questions. And this means that at the end, that's a typo. At the end, it's going to tell me how many I got correct.
So, the idea of this is that you can practice quickly getting the answers from memory so that when you need them in the competition, then you don't have to worry about remembering them.
So, it's an alternative to space repetition software. Uh, so in this case, apparently maybe I should practice my 13 because I forgot that one. And I was slow at answering 12. I think that's just because it's difficult to type. And this one was incorrect. That was a typo.
So, this is a couple of tools you can play with these. Feel free to enjoy those. And that is the end of everything that I have to share today. Thank you very much for joining. Hope you have enjoyed and learned some things during this and yeah good luck for the competition itself.
>> Yes and I'm sure everybody has learned a lot and u as I've mentioned earlier it's all about practice. You have to go ahead and go back and implement all of the learnings over here. You will be having access to the recordings. So go ahead and watch it as many as you as many times as you want and uh and start preparing. As I mentioned, the mock test comes up for the standard mandatory challenge between the 24th and the 27th.
The details will be coming to you on the WhatsApp community groups. If you've not joined it yet, please make sure that you do. It has already been emailed to all of you the WhatsApp community links. If you do not have it, reach out to us on contactiveml.com or drop us in a WhatsApp on plus 918291027238.
All right, with that said, Daniel, thank you so much for your time. It was really awesome to have you here and we look forward to, you know, our collaboration for the future as well. Thank you everyone. You may all now, you know, you know, start logging off and enjoy the remaining of your Sunday. Thank you so much. You really are wonderful and you all deserve the break that you're going to get now. Bye-bye.
She can log off everyone.
Thank you. Bye.
Bye.
by by I7.
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