32.41 $30^{4}$ has, besides 1 and $30^{4}$, how many distinct positive integer divisors?
(A) 100.
(B) 125.
(C) 123.
(D) 30.
(E) None of these.
(25th American High School Mathematics Examination, 1974)
[Solution] Writing 30 as a product of prime factors, $30=2 \cdot 3 \cdot 5$, we get
$$
30^{4}=2^{4} \cdot 3^{4} \cdot 5^{4} \text {. }
$$
The factors of $30^{4}$ are exactly the numbers of the form $2^{i} \cdot 3^{j} \cdot 5^{k}$, where $i, j, k$ are integers satisfying $0 \leqslant i$, $j, k \leqslant 4$.
Thus, $30^{4}$ has $5^{3}=125$ distinct factors, and excluding 1 and $30^{4}$ itself, there are 123 factors. Therefore, the answer is $(C)$.