If $a \mid b$, show that $(- a ) \mid b, ~ a \mid (- b), $ and $ (- a ) \mid (- b)$.

 If $a \mid b$, show that $(- a ) \mid b, ~ a \mid (- b), $ and $ (- a ) \mid (- b)$.

 Answer: Suppose that $a \mid b$. So there exists an integer $c$ such that $b = a \cdot c$.  We can write $b = a \cdot c$ as $b = (- a) \cdot (- c)$, where $(-a), ~ (-c)$ are integers. This gives $(- a ) \mid b$. Now multiply $b = a \cdot c$ by $(-1)$, we will have either $(- b) = a \cdot (- c)$, where $- c$ is an integer or $(- b) = (- a) \cdot c$, where $ c$ is an integer . This gives  $a  \mid (- b)$ or $ (- a ) \mid (- b)$. Therefore we have $(- a ) \mid b, ~ a \mid (- b), $ and $ (- a ) \mid (- b)$.

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