Prove that $\gcd(a,b) = \gcd(b,a)$

Question

I want to prove $\gcd(a,b)=\gcd(b,a)$. I tried using the euclidean algorithm but that didn't help me much.

Answer 1

If $\gcd(a,b)=x$ then $x \mid a$ and $x\mid b$ this implies $x \mid \gcd(b,a)$. By similar argument $\gcd(b,a)\mid\gcd(a,b).$ i.e., $\gcd(a,b) = \gcd(b,a). $

Answer 2

A divisor of $a$ and $b$ is a number $c$ such that $c\mid a$ and $c\mid b$. Note that a divisor of $a$ and $b$ is trivially a divisor of $b$ and $a$.