$\displaystyle

\begin{array}{l}

prove{\rm that} \\

{\rm gcd(a,b)} \le {\rm a - b , a} \succ {\rm b} \\

{\rm solution:} \\

{\rm note}\;{\rm that}\;{\rm gcd(a,b) = gcd(a, - b)} \\

\Rightarrow {\rm d\backslash a , d\backslash - b} \Rightarrow {\rm d\backslash a + - b} \\

{\rm this}\,\;{\rm means}\;{\rm that}\;{\rm d} \le {\rm a - b } \Rightarrow {\rm gcd(a,b)} \le {\rm a - b }{\rm .} \\

\\

\end{array}$