 Hello friends, so welcome again to this session on real numbers and So far we were discussing greatest common divisor or highest common factor If you have not seen our previous videos on GCD and Euclid's division lemma and algorithm I would recommend you to watch them and in this session. I am going to discuss a Typical topic which is you know in studied in parallel with the concept of GCD and that is the concept of Co-prime numbers, so we are going to discuss co-primes Because in number theory, we will come across Lots of cases and examples and concepts where co-primes are dealt with so what is a co-prime? We define two numbers two integers again So it goes without saying that in all these lectures. We are dealing only with integers So always keep that in your mind So let's say if two numbers are there or two integers are there a and b so a and b are co-prime R co-prime co-prime when When GCD of GCD of a and b is equal to 1 Why are they called co-prime? What was the prime number prime number is nothing but number p is prime Again, I'm using the word number. So don't get confused because by default it would mean an integer. So, uh, And especially positive integers. So a number p is prime if if There are There are P is prime if there are only two factors two factors of p factors of P which all one and p itself. So this was a prime example was Example was seven three two itself 11 13 31 all that yeah, there are infinitely many primes. That's another thing where people are You know investigating to find out which is the largest prime you might have come across There's a competition which goes on to find out the largest prime if you are not aware of just Google it and you'll get to know So prime number is an integer which has only two factors one and p now Why are we calling two numbers co-prime? So we mean that a is relatively prime to be and b is relatively prime to a So a and b are prime to each other. That means what there is no common factor between a and b So neither you know apart from one that is So if there are two numbers A and B such that their highest common factor itself is one then they are called Co-primes, so let us take examples of co-prime numbers. So if you see 6 and 11 are co-prime Why because gcd of 6 and 11 is 1 Now you might be thinking that one is prime one of them is a prime number So hence they have to be co-primes But you can have examples where there are two composite numbers yet. They are co-primes example Let's say 9 and 16. So if you if you see both 9 and 16 are not primes They are composite numbers, but if you see gcd of 9 and 16 is one so they are co-prime to each other similarly, if you see 32 and 25 Both are composite numbers, but if you see gcd 32 comma 25 is 1 and To tell you, you know, there is an important observation in case of positive integers and what is that it is this that any two any two consecutive consecutive consecutive integers any two consecutive integers are Co-prime you can check for yourself Examples so one two clearly gcd of one two is one the only one factor that is one gcd of nine and ten both are composite, but still one gcd of 24 and 25 both are composite, but still one so you take any so hence to gen if we generalize it You will say what will you say you will say that? you Will say that gcd gcd of n comma n plus one Right two consecutive numbers is always one where n is a positive Positive integer Actually, it holds true for any negative integer as well But for in number theory mostly we will be concerned with positive integers So hence typically we talk about positive integers, but it doesn't mean that gcd of two negative consecutive integers would not be one Okay, so this is All about co-primes in the next video we will talk about The Euclid's division algorithm and other algorithms to find out the greatest common divisor of any two Positive integers. Thank you for watching this video