Let $a$ and $b$ be positive integers. Prove that $b$ divides $a$ iff $\gcd(a, b)=a$

Question

Having trouble with this euclidean algebra proof!

Let $a$ and $b$ be positive integers. Prove that $b$ divides $a$ iff $\gcd(a, b)=a$

Answer 1

I suppose you mean $gcd(a,b)=b$. Let's denote $g\equiv gcd(a,b)$.

In the forward direction, if $b$ divides $a$, then since $b$ divides itself it is a common divisor, therefore $b \leq g$. But if $b < g$ then we get a contradiction since $g$ needs to divide $b$ so cannot be larger than $b$.

In the reverse direction, if $g=b$, then by definition of $gcd$ we have that $b$ divides $a$.