AMC10 2010 A

AMC10 2010 A · Q9

AMC10 2010 A · Q9. It mainly tests Linear equations, Digit properties (sum of digits, divisibility tests).

A 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，是一个数字反转其各位数字后仍相同的数。数$x$和$x + 32$分别是三位数和四位数的回文数。$x$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

Answer (E): Let $x+32$ be written in the form $CDDC$. Because $x$ has three digits, $1000<x+32<1032$, and so $C=1$ and $D=0$. Hence $x=1001-32=969$, and the sum of the digits of $x$ is $9+6+9=24$.

答案（E）：设将 $x+32$ 写成 $CDDC$ 的形式。因为 $x$ 是三位数，所以 $1000<x+32<1032$，因此 $C=1$ 且 $D=0$。于是 $x=1001-32=969$，并且 $x$ 的各位数字之和为 $9+6+9=24$。

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