Prove that if $a$ and $b$ are coprime then so are $a^n$ and $b^m$

Question

I have to prove that if a and b are relatively prime then so are $a^n$ and $b^m$ by contrapositive I'm asking for help please because i really don't know how to proceed and this assignment is due this afternoon Can someone please just give me a hint or something to start with I tried solving it with different ways but it doesn't work

I'd really appreciate your help.

Answer 1

If $\gcd(a^n,b^m)\neq 1$ then there exist a prime $p$ such that:

$p|a^n,b^m\Rightarrow p|\underbrace{a\times...\times a}_{n}, \underbrace{b\times...\times b}_{m}$

By the definition of prime number :

$p|ab\Rightarrow p|a \vee p|b$

In our case since all of the factors are equal:

$p|a \wedge p|b$

So if $a^n,b^m$ are not coprimes, then $a,b$ are not coprimes

:)