Let $a$,$b$ be relatively prime integers which are both greater than or equal to $2$ . Show that $(a/b)^2$ cannot be an integer.
I have tried to use contradiction to prove this result, but I am not sure that my idea is correct or not. Below is my idea (not a complete proof).
Suppose $(a/b)^2$ be an integer. We have $a/b$ is an integer. Then, we have $a=bk$ where $k$ is an integer since $b$ divides $a$. Since $a,b$ are relatively prime, therefore $b$ must be $1$. Hence, contradiction arises. I am not sure that my idea is correct or not, could anyone help me to check it? If my idea is wrong, could you give me a idea to do this question?