Problem 11 · 2024 Math Kangaroo
Hard
Number Theory
place-valuedivisibilitydigit-sum
A palindrome is a number that reads the same forwards and backwards, for example 121 or 444. What is the sum of the digits of the largest three-digit palindrome that is also a multiple of 6?
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Answer: E — 24
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Hint 1 of 2
A palindrome aba is a multiple of 6 when it is even and its digit sum is divisible by 3.
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Hint 2 of 2
Even means the outer digit a is even; push a as high as possible, then the middle digit.
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Approach: build the largest even palindrome with digit sum divisible by 3
- For aba to be even, a must be even, so the largest choice is a = 8.
- Then 8b8 needs digit sum 16 + b divisible by 3; the largest such b is 8, giving 888.
- 888 is even and divisible by 3, so it is a multiple of 6, with digit sum 24.
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