1) Which of the following numbers is a divisor of $3^{5} \cdot 4^{4} \cdot 5^{3}$?
(A) 42,
(B) 45,
(C) 52,
(D) 85,
(E) 105.
1. The answer is $\mathbf{( B )}$.

We observe that $45=3^{2} \cdot 5$ certainly divides $N=3^{5} \cdot 4^{4} \cdot 5^{3}$. We also note that 42 is not a divisor of $N$ because it is a multiple of 7, while $N$ is not divisible by 7; the same applies to $105=7 \cdot 15$. Similarly: 52 is a multiple of 13, 85 is a multiple of 17, and neither 13 nor 17 divide $N$, because they are prime numbers that do not appear in the prime factorization of $N$.

[Problem proposed by A. Colesanti.]