 Hello friends, so welcome to this session on real numbers. Today we are going to discuss some properties of divisibility. But before that let's do a quick recap on what was divisibility. So if you remember in the last session we discussed that if A divides B and this is how we used to represent you know that A is a factor of B. So what was this? So this statement was nothing but this means A is a factor or a divisor of factor of B. So we say this only when there exists integer C. So mind you here the condition was that A cannot be 0. So division by 0 was not allowed and B can be B is any integer B is an integer. So any integer B positive negative but an integer. So we say that if A is an integer B is an integer but A is not equal to 0 and we say that A divides B only when there exists what as another integer another another integer there exists another integer C another integer C such that B equals A times C. So we learned this in the last session example. So we say 5 divides 20 because there exists a 4 such that 20 is equal to 5 into 4 this is what we learned and we say that 5 doesn't divide 26 because there exists no integer no integer yes there are rational numbers or other real numbers no integers there is a rational number which exists such that you know that particular number when multiplied by 5 gives you 26 but there exists no integer no integer mind you there exists no integer such that 5 into that C that integer will give you 26. So there exists C doesn't exist C doesn't doesn't exist where C is an where C is an integer C is an integer right. So this is what we learned in the last lecture. Now let's go to you know go try and understand some properties of divisibility. So this will be helpful to you while analyzing lots of mathematical problems later on. So let me just tabulate them here one by one it will be a good practice if you also maintain a table like that. So during our childhood or during our school days we used to compile all the results and theorems and you know the other information in one notebook so which we used to refer whenever we used to solve problems. So let's talk about what properties of divisibility of divisibility okay let's see what are the properties of divisibility one by one and we request you to jot down in a separate notebook. So first one is plus one and minus one so plus minus one divides divides every non zero integer. So what I'll do is I'll explain the property and then I'll take examples. So plus one and minus one divides every non zero integer this should not be a big thing to understand because one divides one divides I can write 25 yeah or one divides one divides any a where a is an integer isn't it a is an integer. Similarly minus one also divides 26 or minus one divides any a why because I will by the definition of divisibility for the first case for this case there exists another c which will give me so 25 can be expressed as one into 25 so this obeys the first divisibility definition which we just understood so hence one divides 25 similarly minus one divides 25 this is property number one let's go to the next property property number two okay so property number two what's property number two so second property is zero is zero is divisible divisible by zero is divisible by every non zero integer non zero integer a what does it mean or in mathematical language I can write a divides zero where a is an integer a is an integer isn't it again there doesn't take much of a thought process to understand this because if you see zero can always be expressed as a into another b where b is also an integer b is an integer so again it defines or or it satisfies the definition of divisibility so hence we can say zero is divisible by any every non zero integer a why are we emphasizing this non zero part because we'll see in the third property the third property is divisible or division by zero is not allowed or we say zero does not does not divide an integer does not divide divide any integer so so wherever you encounter a case where there is a division possible by zero then you know that's not allowed so zero doesn't doesn't divide let me just write it properly once again so zero doesn't divide a where a is an integer okay any integer this is property number three so just to you know give a little bit more insight into this so many times what you do is for example you have an expression like a times let's say x minus b is equal to c times x minus b let's say you get an expression like that and we have seen people doing a mistake of cancelling x minus b from both sides it's a classical mistake why because you don't know if x minus b so this basically why can you how can you cancel it because you're saying a by c is equal to x minus b by x minus b this is a usual process now you can cancel this only when you're hundred percent sure that x minus b is not equal to zero so hence you cannot simply cancel this so if x becomes b then this cancellation is not allowed so please be careful so hence zero does not divide any integer so you cannot do an operation like that in fact zero doesn't divide any real number forget about any integer division by zero is not allowed okay property number four property number four is what is property number four property number four is if a is a non-zero integer if a is if a is a non-zero integer again we are talking about non-zero integer non-zero integer non-zero integer and b is and b is any other integer any integer not necessarily non-zero then then then a divides b a divides minus b minus a divides b and minus a divides minus b let's take an example and understand this let's say um uh yeah and and and yes no uh one thing which is missing in this is if a is a non-zero integer and b is any integer then if this is missing and if a divides b then then all these will be implied okay so let's take an example so basically let's say five divides 50 so this automatically means five divides minus 50 how because five into minus 10 will give you minus 50 this similarly minus five divides 50 because again minus five times 10 will give you 50 and then minus five also divides minus 50 why because minus five into um oh sorry in this case it will be minus 10 and in this case it will be minus five into 10 will give you minus 50 so you see if if this holds true if this holds true then all these three will also hold true okay why because it again defines or let's it satisfies the definition of divisibility let's go to property number four okay so we have yeah so let's go to property number four oh sorry four is done so let's talk about property number five