2003 AMC 10B Problem 13

Below is the professionally curated solution for Problem 13 of the 2003 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2003 AMC 10B solutions, or check the answer key.

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Concepts:digits casework systematic listing

Difficulty rating: 1310

13.

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 $\clubsuit(\clubsuit(x))=3?$

$3$

$4$

$6$

$9$

$10$

Solution:

Let $y=\clubsuit(x).$ Since $x\le 99,$ we have $y\le 18,$ so $\clubsuit(y)=3$ forces $y=3$ or $y=12.$

The two-digit numbers with digit sum $3$ are $12, 21, 30$ $(3$ of them $).$ Those with digit sum $12$ are $39, 48, 57, 66, 75, 84, 93$ $(7$ of them $).$ In all there are $10.$

Thus, the correct answer is E.

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