1. The problem statement, all variables and given/known data b^2 = a b is a rational number a is an integer prove that b is an integer. This is self assigned, but I think this is the appropriate place to put my question. 2. Relevant equations see above 3. The attempt at a solution Is this legitimate...? Since b is a rational number, there exists two integers m and n with n=! 0 s.t. b=(m/n) and gcd(m,n)=1, thus we have b^2 = a ; (m/n)^2 = a ; m^2/n^2 = a ; m^2 = a*n^2 ; m = n√a. If a is not a perfect square, then √a is irrational and thus n is irrational since m is an integer...a contradiction. Suppose then that a is a perfect square, then we have that m =nc where c=√a. Since c is an integer, implies that n|m, but since gcd(m,n) = 1 we must have that n = 1 thus b = m and b is an integer. The first part seems a little sketchy to me, I know it to be true...but I'm not sure I can use it how I did. Any advice on how to clean this up? I feel kind of dumb working on my properties of real numbers, but I'm hoping it will help my foundations. Sorry to be asking so many questions, I asked something yesterday as well...maybe I should ask elsewhere in the internet? Though I suppose it is up to the viewer to deem it worthy of response, so I guess it doesn't matter too much as long as I'm not spamming. Thanks in advanced for any help!