 Fair enough folks. Let's go to the next property. Okay property number five it says if a divides B and a is greater than zero and B is greater than zero then Then what happens? Then a is a has to be less than equal to b Okay, a has to be less than equal to b. You can check with an example What's an example? Let's take an example so Example could be let's say a is equal to 8 and b is equal to 24 clearly a divides B as 8 divides 24, but 8 is less than 24, right? So hence this holds This holds. Let's say let's take another one 45 and b is equal to 15 Clearly 45 doesn't divide 15 Right, if it is not dividing there is no possibility of 45, you know Anyways 45 is less than 45 is greater than 15 Correct, so hence if something is dividing some other integer then that divisor has to be lesser than the dividend That's for sure. But how do we prove it mathematically? So let's prove it Let's prove it. So proof is something like that again, we will go by the basic definition of the divisibility So you'll get x as an integer x is an integer right, I hope there is no difficulty in understanding this x is sorry x is x is and Integer, okay, let me write it in a fair manner x is and integer clear So my friends b by a is equal to x right now b by a is an integer and since since a is greater than 0 b is greater than 0 therefore The the quotient b by a has to be greater than 0 and b by a is an integer Isn't it because b by a is x b by a is equal to x is equal to an integer and It is an integer which is greater than 0 guys right greater than 0 Right because b and a both were positive. So that means b by a can be either 1 or 2 or 3 or 4 or 5 or so on and so forth Right, so right so b by a is either 1 2 3 4 or what something like that That means b by a is clearly greater than equal to 1 Right either 1 2 3 4 like that. That means b is greater than equal to a if you multiply the inequality by a positive quantity a from on both sides So you'll get b is greater than 8 so hence proved right Fair enough next is the one more property and this is if m is not equal to 0 not equal to 0 and a divides b then this implies this implies and Is implied by and is implied by and Is implied by What ma is? Factor of mb or m a divides mb now What does this mean as in this means this implies and is implied by what does this mean this mean that If this holds true, then this will hold true where m is an integer m is an integer Okay, m is an integer Mm-hmm. In fact m need not be an integer m could be anything right so m could be anything You need not be an integer where but only criteria is m must not be equal to 0 Right m could be anything right and if this is true then If this is true Then this holds and if this holds Then this is true. This is what is meant by implies and implied by both ways. It is true. That means if a divides b then ma divides mb and If ma divides mb, then you can say a divides b Right clear. This is what it means Let's take an example three divides 21 right, so three into Seven divides 21 into seven, which is quite true Right no problems at all. So four divides 24 and Let's say eight divides 48 So hence this come this means this why because this is four into two Divides 24 into two Quite true. So we are seeing that it is quite true. Let's try to prove it very very simple again. So you can say What will you say here? You will say that a divides b that means b is equal to a x again for some integer x Every proof starts like that Okay, that means I can write I can multiply both sides by a non-zero quantity So mb or bm is equal to ma x or Bm is equal to a m times x Correct, that means a m is equal to or a m not equal to a m divides a m Divides bm Yeah, so this is the one way is proven. Let's say the other way is also true. So let's say let's say if Be a m divides bm Sorry bm if a m divides bm then we can say Bm is equal to a m times some x where x is an integer Isn't it? So we can say bm by m is equal to a x M is not zero so I can cancel m. So hence b is equal to a x Therefore we can declare that a divides b So both ways it's true. That means if a divides b then a m divides bm and If a m divides bm then a divides b. So this is what? The six sub parts of this theorem are so I would recommend that all of you Can just jot them down the results at least at one place So that it becomes easier for you to solve problems and