\(\color{brown}{Link}\) \(\text{Solution:}\) \(Question:\) \(\prod_{i=1}^n \prod_{j=1}^n \frac{lcm(i,j)}{gcd(i,j)}\) 分开得: \[\frac{\prod_{i=1}^n \prod_{j=1}^n ij}{\prod_{i=1}^{n}\prod_{j=1}^{n}gcd(i,j)^{2}} \] 分子即为\((n!)^{2n},\)主要方法在分母. 先不看平方,有: \[\pr…