题意:5e4组询问 给定a,b,d求
\[ \sum_{i=1}^{a}\sum_{j=1}^{b}[gcd(i,j) = d]\]
题解:套路
\[f\left ( d\right ) = \sum_{i=1}^{a}\sum_{j=1}^{b}[gcd(i,j) = d]\]
\[F\left ( n \right )=\sum_{d|n} f(d) = \left \lfloor \frac{a}{n} \right \rfloor\left \lfloor \frac{b}{n} \right \rfloor\]
\[f(d) = \sum_{d|n}\mu(\frac{n}{d})F(n) = \sum_{d|n}\mu(\frac{n}{d})\left \lfloor \frac{a}{n} \right \rfloor \left \lfloor \frac{b}{n} \right \rfloor\]
令t = n/d
\[f(d) = \sum_{t=1}^{min(\left \lfloor \frac{a}{d} \right \rfloor,\left \lfloor \frac{b}{d} \right \rfloor)}\mu(t)\left \lfloor \frac{a}{dt} \right \rfloor\left \lfloor \frac{b}{dt} \right \rfloor\]