I'm reading A Computational Introduction to Number Theory and Algebra, which can be found here as a free download. From the book's exercises, I'm stuck with a proof to show that $\gcd(a,b)=|a| \iff a | b$.
My current reasoning goes as follows: since $\gcd(a,b) = |a|$, we see that $a\mathbb{Z} + b\mathbb{Z}= |a|\mathbb{Z}$.
Since $a|b \iff az = b$, for some $z \in \mathbb{Z}$, we see that both $az, b \in |a|\mathbb{Z}$.
So there exist $s, t \in \mathbb{Z}$ such that $as + bt = az = b$, which is true for $s = 0$ and $t = 1$.
Q.E.D.?
I guess that in the end my reasoning is wrong, but I can't explain it.