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I was going to ask another question about this - Origin - Elementary Number Theory, Jones, p23, Lemma 2.4 - but then I chanced on If a product of relatively prime integers is an $n$th power, then each is an $n$th power and its duplicate. I'm bad at searching and I don't want to instigate more duplicates, thence I want to check if there are any others on this before asking newly.

Is there a name for this Lemma so I can try to search for it here and on Google? If you find anything, please inform me exactly what you typed in.

If $a_1,\dots,a_r$ are mutually coprime positive integers, and $a_1\dots a_r$ is an $m$-th power of some integer $m\ge2$, then each $a_i$ is an $m$-th power.

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Dwayne E. Pouiller
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  • Please just ask the question on the main site. If it's a duplicate, we'll close it, no harm done. – Alex Becker Apr 07 '14 at 07:54
  • @AlexBecker Can you please move this then or do I need to post newly? – Dwayne E. Pouiller Apr 07 '14 at 07:57
  • I'm not aware that there's a specific 'name' for this, but it follows by the fundamental theorem of arithmetic, and also the following property of prime numbers: if $p$ prime is such that $p \mid ab$ then $p \mid a$ or $p \mid b$. – ah11950 Apr 07 '14 at 08:18

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As ah11950 pointed out, this lemma is not prominent enough to yield having it's own name. The best way to refer to it is probably as a "consequence of the Fundamental Theorem of Arithmetic."

Clyde Kertzer
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