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Does $a^n \mid b^n$ imply $a\mid b$? I think it does but haven't been able to prove it. I don't know much number theory so an elementary answer would be great.

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    $\begingroup$ Hint: Look at prime factors. $\endgroup$ – Brett Frankel Apr 5 '13 at 0:29
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    $\begingroup$ You can start with the fundamental theorem of arithmetic. $\endgroup$ – Julien Apr 5 '13 at 0:30
  • $\begingroup$ Consider this. Might I call yhis a duplicate? Since they end up in asking the same question. $\endgroup$ – awllower Apr 5 '13 at 13:32
  • $\begingroup$ @awllower: you're right that it's a duplicate, but I think this one has better answers. $\endgroup$ – Javier Apr 5 '13 at 14:16
  • $\begingroup$ @JavierBadia Probably because the previous one prohibited the use of GCD and UFD. $\endgroup$ – awllower Apr 5 '13 at 15:01
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If you can assume the fundamental theorem of arithmetic (that each integer has a unique factorization in prime numbers), you can write: $$ \begin{align*} a &= p_1^{e_1} p_2^{e_2} \ldots p_r^{e_r} \\ b &= q_1^{d_1} q_2^{d_2} \ldots q_s^{d_s} \end{align*} $$ Here the $p_i$, $q_i$ are primes, and $e_i$ and $d_i$ are all greater than 0. If $a^n \mid b^n$, then $p_i^{n e_i}$ must have a counterpart in a $q_j^{n d_j}$, in that $p_i = q_j$ and $n e_i \le n d_j$, so it must then also be that $e_i \le d_j$; and this means $a \mid b$.

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Hint $\ $ Either examine exponents in unique prime factorizations, or, by the Rational Root Test, the reduced rational root $\rm\:x = b/a\:$ of $\rm\:x^n = c\in\Bbb Z\:$ must be integral, so $\rm\:b/a\in\Bbb Z\:\Rightarrow\:a\mid b.$

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  • $\begingroup$ @Peter For that, after cancelling any common factor, one needs only $\rm\:(a,b)=1\:\Rightarrow\:(a,b^n)=1,\:$ true by iterating Euclid's Lemma. Thus $\rm\:1 < a\nmid b^n,\:$ so $\rm\:a^n\nmid b^n.\ \ $ $\endgroup$ – Math Gems Apr 5 '13 at 0:41
  • $\begingroup$ Yes, that was my idea. I was awfully unclear, sorry. $\endgroup$ – Pedro Tamaroff Apr 5 '13 at 0:44
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Hint: $p$ is a prime factor of $k$ if and only if $p^n$ is a factor of $k^n$. This holds for any prime $p$, integer $k$, and positive integer $n$.

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