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Let $a$, $b$, $c$ $\in\mathbb{Z}$. Prove that if $a\mid b$ and $b\mid c$, then $a\mid (b + c)$.

My proof:

since $a\mid b$, $b = k\cdot a$ for some integer $k$

since $b\mid c, c = l\cdot b$ for some integer $l$

then, $c = l\cdot k\cdot a$, $a\mid c$.

since $a\mid b$, and $a\mid c$, and $b+c$ is an integer combination, $a\mid (b+c)$.

Would it be okay to just write that for my exam and get full marks?

user2345215
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user143251
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  • Looks pretty good. You might formalise it a bit more saying $b + c$ = $l.a + l.k.a$ = $a(l + l.k)$ and therefore $a|(b + c)$. – Tom Collinge Apr 15 '14 at 08:30

1 Answers1

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It may be worth explicitly stating that $b+c=ka + lka=(k+lk)a$, but yes, you're correct.

preferred_anon
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