If $a$ and $b$ are relatively prime integers then prove that ($a$ ,$b^2$) =1 From the title I've stucked in this question for half an hour. Could anyone help me?

Consider the prime factors of $a$ and those of $b$. The fact that $a$ and $b$ are relatively prime means that $a$ and $b$ have no common prime factors. But then $a$ and $b^2$ have no common prime factors either; that is, $a$ and $b^2$ are relatively prime. QED.
(We have used the observation that $b^2$ has the same prime factors as $b$, with each prime factor in $b^2$ repeated twice as many times as in $b$.)
