 hello friends welcome again to yet another session on problem solving on real numbers so in this question the question reads prove that a positive integer n is a prime number if no prime p less than or equal to root n divides n okay in the first reading the question looks like difficult to understand but let us take an example and understand it so the question says that if if a number n is a prime then there will not how to test whether number n is a prime or not so you need to just find out the root of n so let's take an example 99 okay let us check 99 so root 99 is if you see root 99 is 9.94 so this is n root n is 9.94 so what all prime numbers are there less than root n so prime less than root n will in this case will be uh two three five and seven right so we just need to check nine divisibility of 99 by so clearly two doesn't divide 99 but three does divide 99 okay so hence 99 is 99 is a 99 is not a prime right so why am i why are we doing this so if you see we didn't need to check it for prime numbers other prime numbers like 11 or 13 or even 19 17 nothing we just needed to check two three five seven and our job was done another case we'll see let's let's check let's check is n equals to let's say 101 okay 101 so if you see what is 101 root 101 root is root of n is 10.04 okay so what are the factors prime factors less than root n or that is a root less than root uh less than 10.04 is two three five and seven so i just need to check 101 divisibility of 101 by 2357 and i'm done so if you see clearly two is not a factor of 101 because it's an odd number three also is not a factor of 101 because the digits don't add up to multiples of three similarly five doesn't divide 101 for obvious reasons there is no zero or five at the unit's place and similarly 101 is also not divisible by seven so if you see seven also doesn't divide 101 that means none of 2357 divide 101 so we can declare here itself that 101 is prime you can check that no factors even more than seven will be dividing 101 101 and that's what the question is trying to prove so right so this is the question that proves that if you're able to prove the divisibility of any given number uh and divisibility check by prime factors less than root of that number and let's say none of the prime factors divide that particular number then in all certainty that number will be a prime number okay so let us try and prove this i hope this example is understood i'll reemphasize once again so when n was 101 we took root n this is 10.04 and we divide it 101 we check the divisibility of 101 by prime factors less than 10.04 that is 2357 we never cared to divide it by 11, 13, 17, 19 and so on and so forth though these could this could possibly be the factors of 101 but we are content with only these many and if none of these are dividing then in all certainty none no other prime factors will also divide 101 that's what the question is demanding us to prove so let us try and prove that okay okay so let us now try to prove this question so you have to see that uh you have to basically show that n is a prime number if no prime p less than or equal to root n divides n so we'll start with the uh logic and we will say that let n be a let n be a number okay and we are not specifying whether it is a prime or not okay it's a positive integer number that is it is understood that we are talking about it is a positive integer okay such and the condition is let this this be a number such that such that such that no prime no prime less than root n divides n so we can always try and you know we can assume a number n so let n be a number such that no prime less than root n divides n okay and the hence we are rephrasing the question basically now we have to prove that we have to prove that we have to prove that prove that n is a prime number right so what what are we saying let n be a number any number positive integer such that and there's a condition with this number is that there is no prime less than root n that divides n okay so is this this is a number now we have to somehow prove that n is a prime number how do we prove the classical methodology is contradiction method so we are saying let n be a be a composite number composite okay with what with what quality with this quality that no prime less than root n divides n okay okay so n is a composite number with a quality what quality that no prime no again i'm re-emphasizing no prime less than root n divides n okay so now let us say that when n is a composite number so n can always be expressed as two factors p into q isn't it p into q so where we can always assume that one of p one of p is less than root n and sorry one of p and q is less than root n and q is let's say greater than root n isn't it this without loss of generality we can say that if n is equal to p into q so one of the factors will be less than root n and another factor will automatically be more than root n so that this number becomes the product of p and q now let us say now let us let us say let us say that that that a is a prime factor a is a prime factor of p okay so let us say that a is a prime factor of p okay n is equal to p into q where p is less than equal to n root n sorry and q is greater than equal to root n and we are saying a is a prime a is a prime number such that a divides p a divides p so a divides p that means a divides pq okay because now if you multiply q to p also so a is is a factor of pq as well that means a is a factor of n isn't it why because pq was equal to n now from these three statement what can we say we can say that a divides p that means one is less than equal to a less than p okay this is this is one this thing now a is a prime number less than p which is dividing n can you see that a is dividing n so the logic says a is less than p and a is dividing n now also a is less than p and p was less than root n so can we not say that a is less than root n right so we have two conditions here now two things one is a is a is less than equal to root n from here from this logic and from here we are seeing saying a divides n but these two contradicts my first assumption right we what was our assumption our assumption was that a is a composite sorry not a our assumption was n n is a composite number composite number number which doesn't doesn't have a prime factor prime factor less than less than equal to root n now this was our assumption if you see this we started with this assumption only how let us see where was our assumption our assumption was this only isn't it our assumption was that we have to prove that so we assumption was let n be a composite number which what quality this quality that no prime less than root n divides n but now what are we seeing we are seeing that there is one prime factor a which is less than root n and which is dividing n as well so that means our assumption was wrong so hence our hypothesis was wrong so hence we can say hence we can say n is not a composite number because of this only this assumption only all this resulted so hence we can say n is not a composite composite number so if n is not a composite number therefore n is a prime so we just prove that n is a prime number if there is no prime factor p less than root n which divides n okay so I hope you understood this problem and the solution so if you haven't understood anything you can always put that in the comment section and I would urge you to go again and revisit this entire problem solution so that you can understand it properly thanks for watching this video