(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If [itex] 3 | m^2 [/itex] for some integer [itex]m[/itex], then [itex] 3 | m[/itex].

2. Relevant equations

[itex] a | b [/itex] means there exists an integer [itex]c[/itex] such that [itex]b = ca[/itex].

3. The attempt at a solution

I realize that this is a corollary to Euclid's first theorem, and that there are plenty of ways to solve this. However, I need an elementary proof using basic rules of arithmetic and algebra.

[itex]3 | m^2 \quad \iff \quad m^2 = 3k[/itex] for some integer [itex]k[/itex].

So then [itex]3 = \frac{m^2}{k}[/itex].

At this point it seems intuitively obvious that [itex]m[/itex] must be a multiple of 3, but it obviously doesn't qualify as a proof. Could someone give me a nudge in the right direction? Thank you very much!

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# Elementary Number Theory Proof

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