Question

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 $N?$

$2$

$3$

$4$

$5$

$6$

Answer: A

Concepts:palindrome divisibility

Difficulty rating: 1140

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Written solution:

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 Problem 13

Answer: A

Difficulty rating: 1140

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Problem 13 in Other Years