Exercise 0.2.2 ($(k\mid a ext{ and } k \mid b) \Rightarrow k \mid as +bt$)

Question

Prove that if the integer $k$ divides the integers $a$ and $b$ then $k$ divides $as +bt$ for every pair of integers $s$ and $t$.

Richard Ganaye · Accepted Answer

\begin{proof} 
If $k \mid a$ and $k \mid b$, then there are integers $u$ and $v$ such that $a = ku$ and $b = kv$. Then, for all integers $s,\, t$,
$$as + bt = (ku) a + (kv)b = k (ua +vb).$$
If $w = ua + vb$, then $w$ is an integer, and $as +bt = kw$, so $k \mid as +bt$.

\end{proof}

Exercise 0.2.2 ($(k\mid a \text{ and } k \mid b) \Rightarrow k \mid as +bt$)

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Exercise 0.2.2 ($(k\mid a \text{ and } k \mid b) \Rightarrow k \mid as +bt$)

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