Let $a,b \in \mathbb{Z^+},\ a<b,\ d=\gcd(a,b)\ $ and $\ 1<d<a,\ x=\frac ad,\ y=\frac bd,\ x,y \in \mathbb{Z^+}.$
Suppose $a=a_1+a_2,\ b=b_1+b_2,\ a_1<b_1,\ d_1=\gcd(a_1,b_1)$ and $1<d_1<a_1,$
if $\frac{a_1}{d_1}=x\ $ and $\ \frac{b_1}{d_1}=y,\ $ then $\ \gcd(a_2,b_2)=d-d_1$.
It seems this question is very simple,but I dont know if this question is right(include grammar etc.).Could there's a proof for this question?