AMC12 2010 A

AMC12 2010 A · Q6

AMC12 2010 A · Q6. It mainly tests Digit properties (sum of digits, divisibility tests), Base representation.

A $\text{palindrome}$, such as $83438$, is a number that remains the same when its digits are reversed. The numbers $x$ and $x+32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of $x$?

回文数（如 $83438$）是指将其数字倒序后仍保持不变的数。数字 $x$ 和 $x+32$ 分别是三位数和四位数回文数。$x$ 的各位数字之和是多少？

(A) 20 20

(B) 21 21

(D) 23 23

(E) 24 24

Answer

Correct choice: (E)

正确答案：（E）

Solution

$x$ is at most $999$, so $x+32$ is at most $1031$. The minimum value of $x+32$ is $1000$. However, the only palindrome between $1000$ and $1032$ is $1001$, which means that $x+32$ must be $1001$. It follows that $x$ is $969$, so the sum of the digits is $\boxed{\textbf{(E)}\ 24}$.

$x$ 最大为 $999$，所以 $x+32$ 最大为 $1031$。而 $x+32$ 的最小值为 $1000$。但是在 $1000$ 到 $1032$ 之间唯一的回文数是 $1001$，因此 $x+32$ 必为 $1001$。于是 $x=969$，其各位数字之和为 $\boxed{\textbf{(E)}\ 24}$。

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