AMC12 2003 B

AMC12 2003 B · Q8

AMC12 2003 B · Q8. It mainly tests Basic counting (rules of product/sum), Digit properties (sum of digits, divisibility tests).

Let $\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For example, $\clubsuit(8)=8$ and $\clubsuit(123)=1+2+3=6$. For how many two-digit values of $x$ is $\clubsuit(\clubsuit(x))=3$?

设 $\clubsuit(x)$ 表示正整数 $x$ 的数字之和。例如，$\clubsuit(8)=8$ 且 $\clubsuit(123)=1+2+3=6$。有多少个两位数的 $x$ 满足 $\clubsuit(\clubsuit(x))=3$？

(A) 3 3

(B) 4 4

(D) 9 9

(E) 10 10

Answer

Correct choice: (E)

正确答案：（E）

Solution

Let $y=\clubsuit (x)$. Since $x \leq 99$, we have $y \leq 18$. Thus if $\clubsuit (y)=3$, then $y=3$ or $y=12$. The 3 values of $x$ for which $\clubsuit (x)=3$ are 12, 21, and 30, and the 7 values of $x$ for which $\clubsuit (x)=12$ are 39, 48, 57, 66, 75, 84, and 93. There are $\boxed{E}$ = $\boxed{10}$ values in all.

设 $y=\clubsuit (x)$。由于 $x \leq 99$，所以 $y \leq 18$。因此若 $\clubsuit (y)=3$，则 $y=3$ 或 $y=12$。满足 $\clubsuit (x)=3$ 的 $x$ 有 3 个：12、21、30；满足 $\clubsuit (x)=12$ 的 $x$ 有 7 个：39、48、57、66、75、84、93。总共有 $\boxed{E}$ = $\boxed{10}$ 个。

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