AMC10 2003 B

AMC10 2003 B · Q13

AMC10 2003 B · Q13. 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 $x$ 的各位数字之和。例如，$\clubsuit(8) = 8$，$\clubsuit(123) = 1 + 2 + 3 = 6$。有且仅有几个两位数 $x$ 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

(E) Let $y=\clubsuit(x)$. Since $x\le 99$, we have $y\le 18$. 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 10 values in all.

（E）令 $y=\clubsuit(x)$。由于 $x\le 99$，所以 $y\le 18$。因此若 $\clubsuit(y)=3$，则 $y=3$ 或 $y=12$。使得 $\clubsuit(x)=3$ 的 $x$ 有 3 个：12、21、30；使得 $\clubsuit(x)=12$ 的 $x$ 有 7 个：39、48、57、66、75、84、93。总共有 10 个取值。

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