Let ax+by=m, cx+dy=n. Suppose ad>bc. Solving the system we get 0<am-cn<ad-bc 0<dn-bm<ad-bc. Draw the lines 0=ax-by, ad-bc=ax-by etc. They form a rhombus with lattice corners. But the number of solutions is just the number of lattice points inside the rhombus. Applying Pick's theorem we must get the result ( I dind't check it, but I think it's right)

This problem is very similar to following one (from Russian TST'2002): Let $a,b,c,d$ be positive integers such that $ad-bc=k>0$, $\gcd(a,c)=\gcd(b,d)=1$. Prove that there are exactly $k$ pairs $(x,y)\in [0,1)^2$ s.t. both $ax+by$ and $cx+dy$ are integers. [Edited: $\gcd(a,b)=\gcd(c,d)=1$ replaced with $\gcd(a,c)=\gcd(b,d)=1$]

But I cannot> You can proof? Please ! Thanksyou very much!

I don't understand you. iura's solution is absolutely correct, only replace "Draw the lines 0=ax-by, ad-bc=ax-by etc" with "Draw the lines 0=ax-cy, ad-bc=ax-cy etc"