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Let $m ≥ 1$, and let $a$ be an integer. Prove that $a^ 2 \equiv 4\mod 3^m$ if and only if $a \equiv 2\mod 3^m$ or $a \equiv −2\mod 3^m$.
I know that i'm supposed to find $m$ factors $3$ in $a^ 2 − 4 = (a − 2)(a + 2)$, but I don't even know how to get started.
Anyone with tips to prove this?