Problem 13 · 2019 AMC 8
Medium
Number Theory
divisibilitydigit-sum
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?
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Answer: A — 2.
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Hint 1 of 2
List the two-digit palindromes: 11, 22, 33, …, 99. They're all just 11 times something — every one is a multiple of 11. That single fact pins down what N can be.
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Hint 2 of 2
If N is a sum of three multiples of 11, then N is a multiple of 11 too. So scan the 3-digit multiples of 11 in order and grab the first that isn't itself a palindrome — then just confirm it's reachable.
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Approach: every 2-digit palindrome is a multiple of 11
- A two-digit palindrome has equal digits (11, 22, …, 99), which means it equals 11 × (that digit) — always a multiple of 11. A sum of three of them is therefore also a multiple of 11, so N must be a multiple of 11.
- Walk the 3-digit multiples of 11: 110, 121, 132, … The very first, 110, already isn't a palindrome (110 reversed is 011). So N = 110 is the candidate — just check it's actually buildable.
- 110 = 11 + 22 + 77 ✓ three distinct two-digit palindromes, so it qualifies.
- Digit sum of 110 = 1 + 1 + 0 = 2.
- Why this transfers: spotting a hidden divisibility (here, "repeated-digit number = multiple of 11") shrinks an open search into a short ordered list — always ask what number property all the building blocks share.
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