Exercise 1.2.34 ($(a,a+k) \mid k$)

Question

Prove that $(a,a+k) \mid k$ for all integers $a,k$ not both zero.

Richard Ganaye · Accepted Answer

ewcommand{\Z}{\mathbb{Z}}

\begin{proof} Let $d = a \wedge (a + k)$. Then $d \mid a$ and $d \mid a+k$, therefore $d \mid (a+k) -a = k$. So
$$a \wedge (a+k) \mid k.$$
\end{proof}
Note: If we define the gcd by $(a\wedge b) \Z = a \Z + b \Z$, where $a \wedge b \geq 0$, then $\mathrm{gcd}(a,b)$ is defined even if $a = 0$ or $b = 0$, or both, and the property $a \wedge (a+k) \mid k$ is always true, since $0 \mid 0$.

Exercise 1.2.34 ($(a,a+k) \mid k$)

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Exercise 1.2.34 ($(a,a+k) \mid k$)

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