1:(sum_{d|gcd}f(d)=sum_{d=1}f(d)[d|gcd(i,j)])
(这个显然吧。。。)
2:(sum_{i=1}^{n}sum_{j=1}^{m}sum_{d=1}f(d)[d|gcd(i,j)]=sum_{d=1}f(d)sum_{i=1}^{n}sum_{j=1}^{m}[d|gcd(i,j)])
3:(sum_{i=1}^{n}sum_{j=1}^{m}[d|gcd(i,j)]=sum_{i=1}^{lfloor frac{n}{d} floor}sum_{j=1}^{lfloor frac{m}{d} floor}[1|gcd(i,j)]=lfloor frac{n}{d} floorlfloor frac{m}{d} floor)
4：(sum_{i=1}^{n}sum_{j=1}^{m}ij[gcd(i,j)=d]=d^2*sum_{i=1}^{lfloor frac{n}{d} floor}sum_{j=1}^{lfloor frac{m}{d} floor}ij[gcd(i,j)=1])
5:(sum_{i=1}^{n}sum_{j=1}^{m}[gcd(i,j)=1]=sum_{i=1}^{n}sum_{j=1}^{m}sum_{d|gcd(i,j)}mu(d))
（(mu)的性质）
6:(sum_{k|i}mu(frac{i}{k})lfloor frac{n}{i} floorlfloor frac{m}{i} floor=sum_{i=1}^{lfloor frac{n}{k} floor}mu(i)lfloor frac{n}{k*i} floorlfloor frac{m}{k*i} floor)