 Hello friends, welcome to the session on number theory and In this session, we are going to talk about divisibility so since childhood we have been talking about the process of division and you know that a division is a Process of finding a quotient and a remainder. What does that mean? So let's say if you have divided 24 by it right, so you'd have done this method and this is thrice 8 times 3 is 24 and you get a remainder as 0 isn't it? So a remainder is 0 and this happens to be the quotient quotient and this number is called the divisor and My dear friends this number 24 here is called divided This is our Learning isn't it? Now in this particular session, we are going to first define divisibility and Then subsequently we are going to discuss different properties related to divisibility of integers again So we are going to first of all assume that we are going to deal only with integers first of all integers only right and In integers also, we are mostly going to discuss non-negative non-negative Integers that is the whole numbers Right So we will be dealing with different properties of divisibility Okay, so what divisibility is let's first define it and then talk about its properties So let's define divisibility. What is divisibility? So we say that an integer and Integer and every word will become very important. So please mark them an integer B is divisible by divisible by Another integer a or by another non-zero integer Non-zero integer. I will illustrate the meaning of every word But please pay attention to the words which I am writing an integer B is divisible by another non-zero integer a a if There exists if there exists another integer another integer x Okay, another integer x such that such that B is equal to a times x B is equal to a times x Okay, right. This is what is meant by Divisibility, okay, so let's take an example example So we say as we discussed the previous case itself. So 24 is divisible divisible by six or let's say eight in this previous case. We had taken it So 24 is very much divisible by 8. Why because 24 can be expressed as 8 times 3 so here this is my B. This is a And this is x right. So we did find an integer x which is equal to 3 here Which I'm multiplying by 8. I can express 24, right? So let's take another one. So 24 is also divisible by 4 or 6 divisible By 6 y because we again get another integer 4 In this case x is 4 Correct. So this is B a and x. So again, we could find out another integer 4 such that 6 into 4 is 24. So we say that 24 is divisible by 6. Let's take another example So this was first second and third example. Let's take So third example could be let's take a higher number. Maybe 144 Okay, 144 is divisible by 12 Right. Why because we will get another integer 12 times x. I have to find right. So what is that x here? 12 itself. So 12 12 12 times 12 is 144. So we say 144 is divisible by 12. Similarly, all even numbers you can generalize all even numbers All even numbers are divisible by 2 Right, all even numbers are divisible by 2. You can check 4 8 12 24 26 whatever because every time you'll get an x. So 2 into 2 in this case then 8 can be written as um Yeah, it can be written as 4 times 2. So x is 2 here 12 can be written as um uh 6 times 2 Or 2 times 6. So x is 6. Sorry in this case x is 4 Right, and then 24 can be written as 2 times 12. So x is 12 26 can be written as 2 times 13. So x is 13 correct So every even number are every even numbers are or all even numbers are divisible by 2 similarly all Alternate even numbers are divisible by 4 and so on and so forth. You can check all of them Every third number is also divisible by 3. So for example, 3 6 9 12 15 all are divisible by 3 Correct. These are all examples of divisibility. Now if there is something divisible, there can be something not divisible as well for example, so Let's take example 8 is Not divisible by divisible By 3 why because we will not get any integer such that 3 times x x cannot be an integer x is not an integer y Why is that so because if you Try to express x as so 8 is equal to 3 x. So clearly 8 is equal to 8 upon 3 Now this is a fraction or not not. You know, this is not a not an integer Not an integer at all Right. So you can't divide 8 by 3 Isn't it? So not an integer. So 8 is not divisible by 3. Similarly Similarly 36 Is let's say not divisible by 11 Why because 36 into 11. Oh, sorry 36 is equal to 11 times something Correct. Let's say x but x is x cannot be so x is 36 upon 11 Not an integer Right. So x always has to be an integer which we are not able to find out. So hence we say 36 is not divisible by 11 Now we have a special notation for divisibility Notation, what is that notation guys? So let's say if a divides B Right if a a divides b exactly exactly means There is no remainder then we say a and we write this line Bar and then b. So this means a a divides b So what are the meanings of this particular notation a divides b or a is a factor of B Clear examples all the above examples will be handy. So 3 divides 12 6 divides 48 7 divides 21 8 divides 64 and so on and so forth Right. So this is one notation. Please Keep in mind wherever you see such kind of a Language it means a divides b Similarly, if a doesn't divide b then we'll write a This correct. So what does this mean? It means a doesn't divide b Okay, let's take an example again. So 8 doesn't divide 65 7 doesn't divide 59 Doesn't divide 83 and so on and so forth Understood. So that's how this language Yeah, what does this notation particular notation mean? Okay