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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.

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    $\begingroup$ Did you try to use prime factorization? $\endgroup$ – Mark Apr 24 at 10:38
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    $\begingroup$ Do you know that, if $\operatorname{gcd}(a, b) = 1$ and $\operatorname{gcd}(a, c) = 1$, then $\operatorname{gcd}(a, bc) = 1$? You can use this fact and induction to solve this problem. First show $\operatorname{gcd}(a, b^m) = 1$ by induction on $m$, then show $\operatorname{gcd}(a^n, b^m) = 1$ by symmetry of the variables $a, n$ and $b, m$. $\endgroup$ – Theo Bendit Apr 24 at 10:41
  • $\begingroup$ I'll try to solve it this way thank you so much for replying $\endgroup$ – Byun Bacon Gogh Apr 24 at 10:56
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If $\gcd(a^n,b^m)\neq 1$ then there exist a prime $p$ such that:

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

For 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

:)

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  • $\begingroup$ thank you soo much i didn't knew it was simplr you really saved me thank you so much $\endgroup$ – Byun Bacon Gogh Apr 24 at 10:53
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    $\begingroup$ The result you used is known as Euclid's lemma. $\endgroup$ – Chris Custer Apr 24 at 11:54

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