If $a, b, c, d \in \mathbb Z$ and $p$ is a prime factor of $a - b$ and $c - d$, then $p$ is a prime factor of $(a + c) - (b + d)$

Question

By hypothesis, $p \mid (a - b), (c - d)$. Then $p \mid [(a - b) + (c - d)]$. In other words, $p \mid [(a + c) - (b + d)]$.

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