A book asks me to prove a statement but I think it is false The problem below is from Cupillari's Nuts and Bolts of Proofs.

Prove the following statement:
Let $a$ and $b$ be two relatively prime numbers. If there exists an
  $m$ such that $(a/b)^m$ is an integer, then $b=1$.

My question is: Is the statement true?
I believe the statement is false because there exists an $m$ such that $(a/b)^m$ is an integer, and yet $b$ does not have to be $1$. For example, let $m=0$. In this case, $(a/b)^0=1$ is an integer as long as $b \neq 0$.
So I think the statement is false, but I am confused because the solution at the back of the book provides a proof that the statement is true.

This statement is true under certain conditions. You must assume $b \neq 0$. If $(a,b)=1$, you can easily show that $(a^n, b^n) = 1$. This means that $(a/b)^n = a^n / b^n$ is never an integer unless there exists $n$ such that $b^n = \pm 1$, but that means $b = \pm 1$. 
The case where $a = 0$ and $|b| > 1$ must be excluded because then $(a,b) > 1$. 
Hope that helps,
