If $a$ and $b$ are odd, prove $\gcd(a,b) = \gcd(\frac {\left| {a-b} \right |} {2}, b)$ Honestly I don't have a strong idea. I don't know where to even begin, I have considered that the $\gcd(a,b)$ is somehow less than $a-b$, but I'm not even sure why that would be the case.. Any help would be great!

$d|a$ and $d|b$
Iff $d|a-b$ and $d|b$
Iff $d|(a-b)/2$ and $d|b$ (for $d$ is odd)
Same set of common divisors so same gcd
