 Hello and welcome to the session. In this session we discussed the following question which says if q is a prime number then prove that square root q is irrational. Now to prove that square root q is irrational where q is a prime number we will use a result which says that let p be a prime number if p divides a square then p divides a where a is a positive integer. This is the key idea to be used in this question. Now we move on to the solution. We are given that q is a prime number. We need to show that square root q is irrational. We take let square root q be rational then we can write square root q in the form m upon n where m and n are integers having no common factor other than 1 and also n is not equal to 0. Now that we have square root q is equal to m upon n so this means that q is equal to m square upon n square. We got this on squaring both sides so this further gives us q into n square is equal to m square. Now since we know that q divides q n square therefore we have q divides m square since q n square is equal to m square. Now using the result given in the key idea which says that if p is a prime number and p divides a square then p divides a so here we have q is a prime number and q divides n square so this means q divides m since q is a prime number and q divides m square so this means that q divides m so we take let m be equal to p2 for some integer p. Now consider this equation q n square is equal to m square let this be equation 1 so now substituting m is equal to pq in equation 1 we get q n square is equal to p square q square this means we have n square is equal to q p square now since q divides q p square therefore q divides n square since q p square is equal to n square and this implies that q divides n since q is a prime number q divides n square so this implies that q divides n so now we have q divides m and q divides n therefore q is a common factor but this thing that q is a common factor of m and n contradicts the sand n has no common factor other than because earlier we had stated that m and n are integers having no common factor other than 1 thus our assumption that square root q is rational is wrong hence we have that square root q is irrational thus we have proved that square root q is irrational this complete c-session hope you have understood the solutions for this question