 Hello friends, welcome to the session on number theory. We were discussing divisibility in the previous session We understood what divisibility basically meant. How is Divisibility defined now in this particular session. We are going to take up some theorem some properties of divisibility So let's start. Okay. So here we are going to discuss one of the first theorem related to Divisibility now it has some sub parts. What are they first one is this that if a divides B Okay, if a divides B Then then it means or it implies it implies What does it imply? It implies that a also divides BC for some integer C Okay, if a divides B now in all of these we were we are going to assume that a BC all our integers All are Integers so keep that in your mind a BC everything which we are going to discuss here will be related to Integers only so any other constant or any other variable also if you are taking up All of them are integers, right? So if a divides BC for some integer C So let's take first in First let's understand by an example and then we will try to prove this as well. So example So clearly if you take this example that three divides it no three divides. Let's say 24 Three divides 24. So three also divides 24 into two Correct. That is three divides 48 three divides 48 isn't it? So which is true? We can check other one 15 divides 45 So 15 also divides 45 times 4 which is 180 Right, so 15 does divide 180. So this is all you know, this is kind of true Okay, now let's try to prove this how to prove this so a divides B by definition of our Divisibility we can say that there exist an X such that B is equal to a X for again some integer X friends, okay B is equal to a X some integer X now I can multiply both sides by C, right? So if I'm multiplying multiplying both sides multiplying One let us say this is one Multiplying one by C. What will you get? You will get BC is equal to a X C isn't it which I can write as a times X C now X C This thing happens to be an integer, right itself is an integer X C is itself in its integer Y X was integer C was integer So product of two integers will anyways be an integer. So from the statement we get BC is equal to a times X C Right, that means by the definition of divisibility a does divide BC this is property or you know the sub part one of the theorem, right? Let's go to second property second property says or second sub part says that if a divides B if If a divides B and B divides C Okay, so if a divides B and B is divide C then a divides C Then a divides C. Let's take first an example. So let's say five divides 25 and 25 divides 125 Isn't it that means five does divide 125 holds true So let's take another example another example could be seven divides 21 and 21 divides 84 21 times 4 is 84. This clearly means seven divides 84. It does hold Let's try to prove this and prove is again very very easier how to prove that So if a divides B, so we can say B is equal to AX and B divides C. So C can be written as BY where X and Y are integers X and Y are Integers Then what can I not club them together replace? You know B here in the second equation. So let's say this is first Let's say this is second So in the second equation into from two and one or from one and two can I not say What can I say C is equal to B times now B can be written as AX times Y, right? So C is equal to AX times Y. So this can be written as C is equal to AX Y Correct. So that means again from the basic definition of divisibility. We have an integer Which then multiplied by A gets you C. So we can say A divides C. So proved now Let us take the third sub part or third property. Okay, what does it say? It says if A divides B and A divides C Okay, then A divides MB plus NC What is M? What is M and N? M and N are also M and N are integers How do we prove this? So before proving, let's take an example and understand whether this is really true Example, so let's take an example. Let's say three divides 21 right three times seven is 21 and Five divides, let's say 15 Right and let's say M is equal to two and N is equal to three Okay, so two times 21 MB is this if you notice, this is our B This is our A. Sorry. This is C. This one's a this one's a correct. So Two times 21 plus three times 15. Let's find the sum. This is 42 and this is 45 Is it so the total sum is 87? Right and clearly three divides 87 Right, so hence A divides MB plus NC This is validated. Validated right now. Let's try to prove it How do we prove it? Simple again the way we proved the other ones. So by the definition We can say B is equal to A, X and C is equal to A, Y again X and Y are integers X and Y are integers. So that goes without saying Right. So let now let's multiply both sides by MB. So can I not say MB is equal to M times AX and NC is equal to N times AY and let's add the two equations LHS to LHS RHS to RHS. So you'll get A will be common and this will be MX plus NY That means A again divides MB plus NC Why because if you see this item here MX plus NY, this is an integer Is it it? So this is also proved. Okay, let's go to the next one What is the next one guys? Next one says fourth one that if A divides B and B divides A then The only possibility is A is equal to plus or minus B Okay. Yep makes sense. So three divides, you know, so You don't need to take any example to see this. So three either divides three or three divides minus three Okay, so in this case A is either equal to B or A is equal to minus B. All right Okay, so how do we prove it? Let's prove this proof Again by definition very very easy B is equal to AX And A is equal to BY again X and Y are integers Is it? If that is so guys, can I not multiply both the equations together? So you'll get B times A Is equal to AX times BY Isn't it? That means AB in the LHS is equal to AB X Y Since A and B are not zero Because you know in this case A neither A can be zero Nor B can be zero Y because division by zero is not allowed. So in that case what will happen You can just cancel AB from both sides. So you'll get XY is equal to one Right now XY X and Y were integers guys integers, right? And the product of two integer is a positive quantity. That means both X and Y are Greater than zero that is their positive or Both of them are sorry both of them are less than zero together Less than zero together correct, isn't it and And the product is also one. So product of two integers product of two integers Is one when is that possible The only possibility is either of them is one or them or either of them are Negative one in in in any other case. It's not possible So hence we say that either X is equal to Y is equal to one then you will get XY is one or X is equal to Y is equal to minus one then also you'll get XY is one Now in this case if let's say this is true, then what will happen you can say clearly A will be equal to B Why because A was equal to B Y and Y is one. So A is equal to B A is equal to B and if this is true, then what will happen A is equal to B Y. So B times minus one. So hence A is minus B Right. So A is either plus B or minus B. This is what we learned If A divides B as well as B divides A