20) Knowing that between 200 and 300 (inclusive) there are exactly 13 multiples of the integer $n$, what is the value of $n$?
(A) $\leq$
(B)
(C) 8
(D) 9
(E) $\geq 10$
20) The answer is (C).

Let $\alpha n \geq 200$ be the first multiple and $(\alpha+12) n \leq 300$ be the last. To ensure there are no more than 13, we must have $\alpha n - n \leq 300$. The conditions that $n$ must satisfy are then:

- $\alpha n \geq 200$
- $(\alpha-1) n \leq 300$

Subtracting the first inequality from the third gives $12 n \leq 100$, which implies $n \leq 8$.

Subtracting the second inequality from the fourth gives $14 n > 100$, which means $n > 7$.

It follows that $n$ must be equal to 8.

The 13 numbers 200, 208, ..., 296 are all and only the multiples of 8 between 200 and 300 (inclusive).