 Hello and welcome to the session. In this session we discuss the following question which says prove that square root of 5 is an irrational number. Before we move on to the solution let's recall one theorem which says let p be a prime number a square then p divides a positive integer. This is the key idea that we use for this session. Let's proceed with the solution now. Since we need to show that root 5 is an irrational number so we assume let root 5 be a rational number then we say there exists coplying positive integers and b such that root 5 is equal to a upon b where we have b is not equal to 0 then this means we have root 5 into b is equal to a. Now squaring both sides we get 5 b square is equal to a square. This means that 5 divides a square and this 5 is a prime number and according to the theorem stated in the key idea we have that is the prime number p divides a square then p divides a so this means that if 5 divides a square 5 also divides a. Since 5 divides a this means we have a is equal to 5 c for some positive integer c. Now squaring both sides we get a square is equal to 25 c square. Now from here we have a square equal to 5 b square so in place of a square we put 5 b square so we have 5 b square is equal to 25 c square. Since we have a square equal to 5 b square this means we have b square equal to 5 c square or you can say this means that 5 divides b square and here 5 is the prime number so according to the theorem stated in the key idea since 5 divides b square so we get 5 also divides b. So now we have 5 divides a and 5 divides b that is we say since 5 divides a and 5 divides b we find that a and b have at least 5 as a common factor we had assumed a and b to be the co-prime positive integers so we have but a and b this is a contradiction our assumption is wrong that we have that root 5 is an irrational number. Now since we had assumed root 5 to be a rational number and we have proved that our assumption is wrong that we get that root 5 is an irrational number. So this completes the session hope you have understood the solution of this question.