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Is there an automated way to prove really boring elementary number theoretic results?

Motivation: I'm writing a proof, and within it, I need to prove: Conjecture: Let $p$ be an odd prime (i.e. $p \neq 2$). Let $c \geq 2$, $d \geq 1$ and $r \geq 1$. If $p^r$ divides $cd$, then ...