Question

回文数是从左到右读和从右到左读数值相同的数。（例如一万二千三百二十一是回文数。）设 $N$ 是最小的三位整数，它不是回文数，但可以表示成三个不同的两位回文数之和。 $N$ 的数字和是多少？

A palindrome is a number that has the same value when read from left to right or from right to left. (For example, 12321 is a palindrome.) Let $N$ be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of $N?$

$2$

$3$

$4$

$5$

$6$

答案：A

知识点：回文数整除性

难度评级：1140

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文字解答：

$2$ 位回文数的十位和个位相同，所以都是 $11$ 的倍数。

如果一个 $3$ 位数是 $3$ 个不同的 $2$ 位回文数之和，那么它也必须是 $11$ 的倍数。

最小的 $3$ 位 $11$ 的倍数且不是回文数的是 $110$ 。它可以写成 $11 + 22 + 77$ 。

因此 $N = 110$ ，数字和为 $1 + 1 + 0 = 2$ 。

正确答案是 A。

Note that $2$ -digit palindromes have the tens and units digits the same. This means that they are all multiples of $11.$

If a $3$ -digit number is the sum of $3$ $2$ -digit palindromes, then it itself must also be a multiple of $11.$

The smallest $3$ -digit multiple of $11$ that is not a palindrome is $110.$ This can be achieved by adding $11 + 22 + 77.$

Therefore, $N = 110.$ The sum of its digits is $1 + 1 + 0 = 2.$

Thus, the correct answer is A.

2019 AMC 8 第 13 题

答案：A

难度评级：1140

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文字解答：

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