The Remainder is Prime or 1

追加: 2026-05-23

[00:00:02]Welcome to another video.

[00:00:05]We know that prime numbers are not divisible by any other number except themselves and one.

[00:00:13]So, if we take any prime number and divide it by 30, the task in this video is to show that the remainder will be one or a prime number. No other options.

[00:00:27]Let's get into the video.

[00:00:30]>> [music] [music] >> So, the first thing that we want to recognize is that whenever you split a number into two, you have to look out for the properties of the split. Let me tell you what that means. This is the most important fact in this problem. And in fact, it is the only fact that we need. Anytime you split a number like this, if A can be split into B + C, if any number divides both B and C, then that number must be the divide A. Or, if any number divides A and B, then that number must divide C. So, if any two of them have a common factor, it is also a factor of the third. That's another way of stating this. If D is a factor of any two, then D is a factor of the third. It doesn't matter which two something can divide. Let me give you a very clean example.

[00:01:35]Let's say we split 20 into 14 and 6. Notice that the only number that divides 14 and 6 is 2. So, that number must also divide 20.

[00:01:50]Okay?

[00:01:52]Now, notice this. Let's take a prime number, 11.

[00:01:57]There is no way to split 11 into two numbers so that there's a number that divides both of them. It's impossible.

[00:02:06]Okay? Because if a number divides any the two of them, then that number has to divide 11. Well, there is no number that divides 11 except one. And that's the whole concept in this question. There's something unique about 30 though. It's because of what it carries, okay? Um and that's what we're doing. So, we're going to say if a prime number is divided by 30 has remainder remainder R then we can write the prime number as a multiple of 30 plus the remainder. How many times will 30 divide the prime number? K times, okay? What's the remainder? R. What we need to show is that because 30k and R cannot have a common factor because their common factor will have to divide P, but that common factor cannot unless it's one, right? So, these must be relatively prime.

[00:03:14]So, no number that divides 30k can divide R. And that's it. So, the question is what could R be? What are the possible remainders when you divide a prime number by 30? Well, the possible remainders are all numbers less than 30, right? So, possible values of R will be 1 2 So, notice that all the possible remainders are these. We can't include zero because if you say it's zero, then it means 30 divides a prime number, which we know doesn't make any sense.

[00:03:58]Okay, and we can't write 30 for R because 30 is 30, which will make it divisible also. So, here one thing one thing we know is that anything that divides 30 cannot divide R, and these are the possible [clears throat] values of R. Okay? The prime factors of 30 are two, three, and five. Actually, 2 * 3 * 5 is 30. So, those three prime numbers cannot divide any number here. If it divides it, it's going to break this rule that we just made. So, all we got to do is eliminate all even numbers.

[00:04:38]Eliminate all Eliminate all multiples of three.

[00:04:50]>> [sighs and snorts] >> Then eliminate all multiples of five.

[00:04:56]This is gone.

[00:05:00]This is gone.

[00:05:04]What is left?

[00:05:06]Now, see what's left on the board.

[00:05:09]Everything on the board is a prime number plus one.

[00:05:16]And that's the proof.

[00:05:22]So, since the prime factors of 30 are two, three, and five, we must eliminate all multiples of two, three, and five.

[00:05:30]That we just did, and the only numbers left are one and all the prime numbers less than 30.

[00:05:38]So, so, R can only be one.

[00:05:50]7 11 13 17, 19 23 29 Did I miss any number?

[00:06:05]1 7 11 13 17 19 23 and 29 Therefore P over 30 has remainder that is prime or 1 Nice little proof there. Never stop learning. Those who stop learning stop living. Bye-bye.