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If $\gcd(a,b)=1$, then $\gcd(a^n,b^n)=1$

This seems clear, but I don't know how to prove this..

I was trying to show this by induction such that if $a^{n+1}$ = $rs$ and $b^{n+1}$ = $rt$, then $s,t$ are divisible by $a,b$ respectively, but i think this is a wrong way..

  • 5
    Suppose prime $p\mid a^n$ and $p\mid b^n$, thus $p\mid a$ and $p\mid b$, so $p\mid\gcd(a,b)$.2012-07-05
  • 1
    Induction is the way to go here! Prove by induction that $(a^n,b)=1$.Then that $(a,b^n)=1$. Profit.2012-07-05
  • 0
    Possible duplicate of http://math.stackexchange.com/questions/63323/how-to-use-fundamental-theorem-of-arithmetic-to-conclude-that-gcdak-bn-12012-07-05
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    @lhf Not exactly a duplicate since that question specifically asks for the proof using fundamental theorem of arithmetic.2012-07-05

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