Is my proof correct? Let $a, b, c\in\mathbb Z$. Prove that if $a\mid b$ and $b\mid c$, then $a\mid(b + c)$.

Question

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?

Answer 1

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