17.2.1 * How many positive integers $n$ are there, which are divisors of at least one of the numbers $10^{40}, 20^{30}$?
Parse $10^{40}, 20^{30}$ whose greatest common divisor is $2^{40} \cdot 5^{30}$. Therefore, the number sought is $d\left(2^{40} \cdot 5^{40}\right)+d\left(2^{60} \cdot 5^{30}\right)-d\left(2^{40} \cdot 5^{30}\right)=41^{2}+61 \cdot 31-41 \cdot 31=2301$.
Note: Generally, the number of positive integers $n$ that satisfy $n \mid a$ or $n \mid b$ is $d(a)+d(b)-d((a, b))$.