CAT Infinite DILR - Set 467 | The Fruity Subtraction | Cryptarithmetic

[00:00:00]Hello everyone. Welcome to Aptitude Jab.

[00:00:03]This is a set based on cryptarithmetic.

[00:00:07]And we are given a subtraction over here.

[00:00:10]So it says that in this cryptarithmetic problem, each distinct letter represents a different digit. Okay, so A will represent a different digit, E will have a different digit, N will have a different digit, and so on. And identical letters represent the same digit. So this digit will be equal to this digit, will be equal to this digit.

[00:00:31]Okay?

[00:00:32]No leading digit is zero.

[00:00:35]These are some unsaid rules, by the way.

[00:00:37]Leading digit is the first digit from the left side. So they cannot be zero.

[00:00:42]So B cannot be zero, A cannot be zero, L cannot be zero.

[00:00:47]Now, subtraction is a bit tricky one to solve. So what we will do is we will convert into addition, because uh carry is easy to understand than the concept of borrow. So what we can do is we can say apple plus lemon is equal to banana.

[00:01:16]And we will solve it now, and that would be a bit simpler to solve. And at the same time, we will make a table like this, giving digits to different alphabets.

[00:01:32]Now, two very basic rules of cryptarithmetic addition, that when we add two two numbers having the same number of digits and we get an extra digit, that is always one. Okay?

[00:01:44]So uh like when we add two numbers, let's say we add 2 + 3, we get five, right? If you're adding a larger number, let us say we add 7 + 9, we are getting 16, right? So what do see, There is a carry.

[00:01:58]So, the concept is carry is equal to zero or one.

[00:02:03]So, if there is an extra digit, that means there is a carry of one. So, B is one.

[00:02:09]We will also make a table list here wherein we will put the numbers and get the sum.

[00:02:20]This digit is one.

[00:02:22]Now, next important rule of cryptarithmetic addition is this. A plus L is equal to A.

[00:02:28]In this scenario, either L is equal to zero or L is equal to nine. So, let us understand that.

[00:02:35]What if we put L equal to zero?

[00:02:38]Okay. So, if you put L equal to zero, it should be A. But, in that case, there will no be no carry. Let us understand through certain example. If you put seven, seven plus zero is seven, but how can we get a carry in that case?

[00:02:52]So, we will take the other case, which is L is equal to nine and there is a carry from behind.

[00:02:58]So, if there is a carry from here and L is equal to nine, then we will get A.

[00:03:03]Suppose A is four.

[00:03:05]So, one plus four plus nine, that is 14.

[00:03:08]Okay. So, 14 and there is a carry over here, which is in the form of B.

[00:03:14]So, that means L is nine.

[00:03:17]Let us put nine in these two places.

[00:03:21]Now, the entire thing is about carry.

[00:03:22]So, whether there is a carry or not no carry.

[00:03:25]So, L is nine that we have figured out.

[00:03:28]Now, uh few more things.

[00:03:32]This thing is E plus N is equal to A.

[00:03:36]Now, if E plus N is greater than 10, in that case, there will be a carry.

[00:03:46]That carry will reflect here.

[00:03:49]And we know L is is to nine.

[00:03:52]So, if you take any digit for O, let us say if we take five here, so five plus 10 15, so N it will be the same digit.

[00:04:00]Right?

[00:04:01]So, it means or if you take any number, eight then it will be 18, so eight but O and N are different digits, so that means there is no carry.

[00:04:12]Okay, there is no carry.

[00:04:15]Now, no carry, so it means E plus N A is equal to E plus N and it is less than 10.

[00:04:22]Right?

[00:04:24]Or you can also say that it is less than nine because we do not have nine available.

[00:04:30]Nine is already taken by L. Right? Now, second thing which is again coming from this execution itself L plus O if you do, you will get N.

[00:04:41]Right? With one carry over here. There is a carry this side which goes here.

[00:04:46]And if you take O any digit like we took eight and we got seven.

[00:04:50]If you take O as five, you will get four.

[00:04:52]So, another observation will tell you that O is equal to N plus one. Okay, so these are two relations that we got from the rightmost digit. Okay, so there is no carry here. Zero carries.

[00:05:06]Now, let us figure out what could be the possible values of E and N A. Right? So, we will have to make cases over here.

[00:05:14]Now, first thing is E plus N is uh minimum values E and N can take is two or three because uh they cannot be zero. Okay, why they cannot be zero? If you take zero here, so zero plus N will be N which is not possible. Right? If you take zero here, E plus zero will be equal to E which is again not possible.

[00:05:39]So, that is why none of them can be zero. Okay?

[00:05:43]One is not possible because one is already taken by B.

[00:05:48]So, minimum if you take two and three in any order, that will be five. So, A could be five.

[00:05:55]And maximum is eight because A is a single digit number.

[00:06:00]And A is less than nine. So, A is ranging from five to eight. Five, six, seven, eight.

[00:06:09]Right?

[00:06:10]Now, we will have to make some cases.

[00:06:13]Suppose if we take A is equal to five.

[00:06:17]Right? Then what are the possible values of E and N?

[00:06:21]Now, one thing to consider here is another thing which which is very crucial here is if you see this column.

[00:06:29]B plus E gives us N. Right?

[00:06:33]Now, if we consider and we have to generate a carry also. Notice this thing very carefully that we have to generate a carry also.

[00:06:43]So, what does that mean? That this N is accompanied by a one over here. Then only we will get a carry. Right?

[00:06:51]Now, if you add any digit to E, like we are adding two digits and we are getting N. So, understand through this example.

[00:06:59]If I make 7 + 5, we get 12.

[00:07:02]If I make 9 + 6, we get 15. Right? Any two digits if you add, so this digit will always be less than the two digits over here.

[00:07:15]This digit is less than two digits over here. Then only we will get a carry.

[00:07:20]Right? If this digit is higher, we will not get a carry.

[00:07:23]Take any any other example also.

[00:07:26]Wherever you have to generate carry, let us say if you do 6 + 5, that is 11.

[00:07:30]Right? So, this is digit is smaller than this.

[00:07:34]So, it means N will be smaller than E.

[00:07:40]Okay? Otherwise, the carry will not be generated. Right? So, N is less than E.

[00:07:45]Now, N is less than E. So, can we take two and three?

[00:07:50]N is less than E, right?

[00:07:53]Uh if you take two and three, in that case, O becomes three.

[00:07:59]Okay, N plus one. So, if you take N is equal to two, E is three, so O becomes three, which is not possible because two all represent different digits. So, A is equal to five is not possible. Okay?

[00:08:12]Let's try with A is equal to six, E is equal to seven, E is equal to eight.

[00:08:18]Now, in this case, we have to take E is uh N should be less than E, O should be N plus one.

[00:08:26]So, we cannot take five and one.

[00:08:29]Okay, or we can say that E has to be greater than N. That's how we will treat. Five and one is not possible because if you take five, B is already one. So, one possibility is four and two here.

[00:08:41]If we take seven, six one is not possible, right?

[00:08:45]Five two is possible.

[00:08:47]And we cannot take four three because O is N plus one. So, O will also be four.

[00:08:53]Right? So, that is one possibility. If we take eight, eight means we will have to take six two.

[00:09:01]Or we will have to take five three. So, these are the four cases that you will get for the possible values. Okay? So, uh try each of these cases, okay?

[00:09:14]Logically, uh we we try with the flow, right? We start with five, six, seven, eight, right? Sometimes, uh it is luck-based thing also that if you are trying from five, six, seven, eight, and you end up getting the result faster.

[00:09:30]Okay, so let us try with six. Let us put four and two.

[00:09:34]So, if you put E is equal to four and N is equal to two.

[00:09:38]So, you get six over here.

[00:09:40]Okay. Now, this is three we know. So, we have l is nine, nine plus three 12. So, let us do the calculation this side so that we can put the numbers later. 4 + 2 is 6 and then 9 + 3 is 12. We get a carry over here. P is missing, m is missing.

[00:10:00]P is missing, e is four that we put here and n is two, right? And l is nine and a is six by this logic. So, 6 + 9 and we are saying that there is a carry generated over here. So, six is this and here also we have six. Right? So, these are the digits that we are obtaining. Now, we need to get a value of two and we have to generate a carry also, right? So, one thing that is possible is 8 + 4 is 12.

[00:10:31]And that is what we get, 8 + 4 12.

[00:10:34]So, we will if we take l p is equal to eight. Right?

[00:10:39]So, 8 + 4 is 12 and we will get a carry over here.

[00:10:42]Now, if you put p is equal to eight, okay?

[00:10:46]Now, if you see this, we are taking eight as here. So, if you take eight eight, 8 + 1 is 9, 9 + 7 is 16, but we will get a carry over here. Like if you take p is equal to eight, we are getting 8 + 7 16 8 + 7 15 + 1 16 and a carry here, which is not giving two. So, we cannot take eight here.

[00:11:08]Another possibility is we have a carry and we have p as seven here. Now, if p is seven, we will have this equation and this is 7 + 4 11 12, one carry from this side. So, 8 + 8 gives us 16. So, if you see this, this matches completely. Okay, so just re-look at the sum. So, nine 4 + 2 is 6, 9 + 3 is 12, 1 carries.

[00:11:36]1 + 7 + 8 that is 16, 1 carries. Okay, and then we have 7 + 4 is 11.

[00:11:43]12, 1 carries.

[00:11:45]15, 16, and this is the result. And so, we got lucky in using this relation. You can try with these three and see if you get the answer or not. Ideally, these problems have unique solution. So, in the examination also, if you are trying cases, if you get lucky in a particular case, just go with it. Do not try multiple cases, unless there are some questions that say cannot be determined.

[00:12:10]Which digit does L represent? L is 9, that we figured out early. So, even if you attempted this part and left, you could have solved one question in much lesser time.

[00:12:21]Which letter does P represent? So, P is 7 over here.

[00:12:26]What is the value of lemon? So, lemon is uh we used below, that is 13.

[00:12:32]13 + 10, 23, 26.

[00:12:37]PAN - BOE. PAN is uh P is 7.

[00:12:43]A N A N A N are 6 and 2, 762 - BOE.

[00:12:51]B is 1.

[00:12:53]O is 2.

[00:12:56]No, O is 3.

[00:12:58]And E is E is 4.

[00:13:05]So, we will get 628.

[00:13:11]So, this was the solution to the set and the answers to the questions.