1986 AIME 第 8 题

Factor $1000000$ and count all of its positive divisors

Pair every divisor $d$ with its complementary divisor $\frac{1000000}{d}$

Let $N=1000000=2^6 5^6.$ It has $(6+1)(6+1)=49$ positive divisors. The product of all of them is $N^{\frac{49}{2}},$ so the sum of their base-$10$ logarithms is $$\frac{49}{2}\log_{10}N=\frac{49}{2}\cdot6=147.$$ The proper divisors include $1$ but exclude $N$ itself. Subtracting $\log_{10}N=6$ gives $S=141,$ already an integer.