 So, let's talk about property number five. So, what's property number five? Property number five says that if A and B are non-zero integers. What does it mean? A and B are non-zero integers and let's say A divides B and B also divides A. Let's say this is there. Then this can happen only when A is equal to plus minus B. Okay? So, again we can take an example and understand. A divides B. Let's say seven divides seven. Okay? And, and, and so let's say this is A and this is B. So, seven clearly divides B. Obviously, this, this seven divides. So, let's say this was A, this was A and this was B. So, in this case this B goes here and this A comes here. So, this is true. This is, so hence we see that A is equal to plus B. So, this is one case. Another case is when let's say seven divides minus seven and, and minus seven divides seven. Second case number two. Again in this case this was my A, this was B and here this is B and this is A. Correct? So, if you see this holds true only when A is equal to plus minus B. So, otherwise there is no other possibility. So, you can take another example which you know where A and B are not same. Let's say seven divides fourteen. Okay? So, seven divides fourteen. So, let's say this is A and this is B. Yeah? But under no circumstances we can say that fourteen divides seven. Yeah? It is not true. Why? Because we don't, we don't have any integer C such that seven is equal to fourteen times C by the definition of divisibility. So, hence there exists this is not an integer. Not an integer. Not an integer. So, it doesn't hold. Okay? This was property number five. Please remember these properties. Now, property number six. Property number six says, let's talk about property number six. Property number six says, if A is non-zero integer property number six. If A is non-zero integer. See, in every property this has been highlighted. So, please be very careful. If A is non-zero integer and B and C are, B and C are any two integers. Any two integers and again one point to be highlighted is we are talking only with only about integers. Right? Then, then if, if A divides B and it's given that A divides C. Okay? It's given. It's given. Then, then we know that first property is, first property is A divides, let me just enumerate it. So, A divides first of all B plus C. Okay? Second is A divides B minus C. Third is, third is A divides B times C and fourth is A divides B times X for, for any other integer X, any the integer X. So, these are, these are very vital properties. Now, let us take examples and understand. So, let's say A is equal to three. So, and B is equal to six and C is equal to twelve. Okay? So, clearly A divides B. Yeah? Three divides six. Clearly A divides C also because three times four is twelve. So, A divides B and A divides C holds. Now, let's check. What is B plus C? B plus C is equal to six plus twelve, which is eighteen. So, clearly, clearly A divides B plus C. Okay? Let's find out B minus C. So, this is property number one. Validated property number two. B minus C is six minus twelve minus six. Clear? So, if you see A, again three divides minus six. Clear? Why because three divides minus six? Because three into minus two will give you minus six. So, two is also validated. Point number three, let's find out B into C. B into C is six into twelve is seventy-two. So, clearly A divides B C. That is, this implies three divides seventy-two. Clearly, is it? Three times twenty-four is seventy-two. So, three divides seventy-two. This is also, looks like true. Fourth is, let's say X is five. So, let's say X is, let's say X equals to five. So, BX will be six into five. That is thirty. Okay? So, clearly A divides BX as well. Yeah, because three divides thirty. That means three into ten is thirty. So, point is all those are true. But how do we prove that? How do we prove that? So, proving is also not that difficult. So, let's, you know, I'll do one proof for you guys and then you can treat, take others as practice. So, let's say, if A divides B, if A divides B, so I can say B is equal to, B is equal to K times A. Let's say K1 times A, where K1 is an integer. K1 is an integer by our definition of divisibility. Now, similarly, A divides C. So, I can say C is equal to K2 times A for another K2, which is an integer. K2 is an integer. Isn't it? So, now, let's say B plus C. What is B plus C? B plus C will be K1A plus K2A, which can be written as K1 plus K2 times A. Now, if you see K1 plus K2, if K1 is an integer, K2 is an integer. So, this sum of two integers always is an integer. So, this is an integer, let's say K. So, hence, I can say B plus C is equal to K into A. That means, what do I know? A divides B plus C. So, this is the proof. So, you guys can try the similar proving for all the other properties. So, let's go to property number 7. So, property number 7 will do here. So, here, we are going to discuss property number 7. What is property number 7? 7 is, yeah, property number 7. So, I hope you guys are jotting down. If not, then you can pause the video and always jot down the property. So, property number 7 says, if A and C are non-zero integers, if A and C are non-zero integers, and B and D are any two integers, and B and D are any two integers, integers, integers, then property number 1, 7, 1 is if A divides B and C divides D, this implies A C divides BD. This is property number 1, 7, 1, and 7, 2 is A C divides B C. A C divides B C. This means A divides B. So, let's take an example and then we'll try to validate it. So, let's take this one first. So, A divides B and C divides B. So, let's say 3 divides 12 and 5 divides, let's say 25. So, what is A? So, A is 3, B is 12, C is 5, and D is 25. So, our two conditions hold. So, now let's find out A C. So, A C is nothing but 3 into 5 is 15. And B D is 12 into 25, which is 300. So, if you see clearly 15, which is A C divides 300. This is A C, this is B D. So, 15 into 20, 15 into 20 will give you 300. So, hence 15 divides 300. So, this looks like valid, but this is not approved. So, you have to prove it out. So, let's after discussing point number 2 and then we'll see proves. So, now, again, let's say A C divides B C. So, let's take an example. So, in this case itself, let's see 3 into 3 into, let's say A C was 15 and B C is 12. So, let's say 3 into 5. 3 into 5 clearly divides B. So, B was 12, 12 into 5. 3 into 5 divides. If this is true, is it true? Yes, it is true. How? 15 divides 60. Why? 12 into 5 is 60. 3 into 5 is 15. So, clearly 15 divides 60. How? 15 times 4 will give you 60. So, 15 divides 60 clearly. So, hence, we say that 3 also will divide 12, which is also true. So, hence, both these properties seem to be true. But let's try and understand the proof of it. So, let's try the proof of point number 1, point or property number 7, 1. So, what is the proof? So, let us see here. Now, so, it's said that proof of, so, let's try, we are trying to prove, prove of what? 7, 1. And then 7, 2. Again, you can try at home. If not, then you can always put it as a comment. Okay. A divides B. So, by our definition, I can say B is equal to K1A. Again, where K1 is an integer. Similarly, I can say D is equal to K2C, where K2 is also an integer. By our definition of divisibility, we just learned this. Now, multiply both of them. So, hence, I can say B, A, yeah, let's, this is 1. Let's just say this is 1 and this is 2. So, we can multiply the two equations and say left hand to left hand and right hand to right hand. So, B into D will be equal to K1 into A times K2 into C. So, then BD is equal to, I can write K1K2 times AC. So, clearly, this particular term is an integer. Why? Product of two integer is always an integer. Okay. That means, from the divisibility definition, I can say, I can say what can I say AC divides BD. Okay. So, hence, proofed. Yeah. Similarly, you can try the other proof. So, these are the properties regarding divisibility. There are a few more which will take up in the other session. These, the other properties will be a little advanced properties which are normally used in mathematical contest like mathematical Olympiad. So, this knowledge is good enough for J, I'm sorry, board level preparation for the other, let's say higher level stuff. We'll have another session on that. Thanks a lot for watching this video.