A corollary of greatest common divisor and division. Say you have some b|a such that:
\begin{equation} a = bq + r \end{equation}
Now, d|a,b \Leftrightarrow d|b,r (because d|b,r implies there’s some x, x’ such that a = (dx)q+dx’, and so a = d(xq + x’) and so d|a; the logic goes the other way too). This finally implies that \gcd (a,b)= \gcd (b,r) because any divisor that works for one works for both.