Prove or disprove. Let 𝑎,𝑏, and 𝑑 be integers with 𝑑≠0. If 𝑑 | 𝑎 and 𝑑 | 𝑏, then 𝑑2

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Question 3

Prove or disprove. Let $a,\ b,$ and $d$ be integers with $d \ne 0$ . If $d \mid a$ and $d \mid b$ , then $d^2 \mid ab$ .

Assume $d \in \mathbb{Z} \setminus \{0\}$ .

If $d \mid a$ and $d \mid b$ , then, there exists $q,\ r \in \mathbb{Z}$ such that $dq = a,\ dr = b$ .

Then, $ab = (dq)(dr) = d^2(qr)$ .

Since $qr \in \mathbb{Z}$ , $d^2(qr) = ab \Leftrightarrow d^2 \mid ab$ .

Thus, $d^2 \mid ab$ .

■ (Q.E.D)

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