A1. A divisor of a natural number $n$ is a proper divisor if it is different from 1 and $n$. For how many natural numbers $n$ is the number 17 the greatest proper divisor?
(A) 0
(B) 1
(C) 3
(D) 5
(E) 7
A1. Since 17 is a proper divisor of the number $n$, we have $n=17 k$ for some natural number $k \geq 2$. Since $k$ is also a proper divisor of the number $n$, it must be that $k \leq 17$. At the same time, $k$ must be a prime number, because if $k=ab$ for some natural numbers $a, b>1$, then $17a$ would be a proper divisor of the number $n$ that is greater than 17. Therefore, $k \in \{2,3,5,7,11,13,17\}$ and all these possibilities are valid. There are 7 such natural numbers $n$.