AMC8 2008

AMC8 2008 · Q22

AMC8 2008 · Q22. It mainly tests Fractions, Divisibility & factors.

For how many positive integer values of $n$ are both $\frac{n}{3}$ and $3n$ three-digit whole numbers?

有且仅有几个正整数 $n$ 使得 $\frac{n}{3}$ 和 $3n$ 都是三位数的整数？

(A) 12 12

(B) 21 21

(D) 33 33

(E) 34 34

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): Because $\frac{n}{3}$ is at least 100 and is an integer, $n$ is at least 300 and is a multiple of 3. Because $3n$ is at most 999, $n$ is at most 333. The possible values of $n$ are 300, 303, 306, ..., 333 = $3 \cdot 100, 3 \cdot 101, 3 \cdot 102, ..., 3 \cdot 111$, so the number of possible values is $111 - 100 + 1 = 12$.

答案（A）：因为 $\frac{n}{3}$ 至少为 100 且为整数，所以 $n$ 至少为 300 且是 3 的倍数。因为 $3n$ 至多为 999，所以 $n$ 至多为 333。$n$ 的可能取值为 300、303、306、…、333，即 $3 \cdot 100, 3 \cdot 101, 3 \cdot 102, …, 3 \cdot 111$，因此可能取值的个数为 $111 - 100 + 1 = 12$。

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