Problem 6 · 2025 AMC 8
Medium
Number Theory
divisibilitymod-10
Sekou writes down the numbers 15, 16, 17, 18, 19. After he erases one of his numbers, the sum of the remaining four numbers is a multiple of 4. Which number did he erase?
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Answer: C — 17.
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Hint 1 of 2
You don't need the actual sums — only the leftover after dividing by 4 matters. What's each number's leftover?
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Hint 2 of 2
Find the leftover (remainder) of the total when divided by 4. The number you erase must carry away exactly that much leftover.
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Approach: track only the remainders (leftovers) mod 4
- The full sum 15+16+17+18+19 = 85, and 85 = 84 + 1 leaves a leftover of 1 after dividing by 4. To make the remaining four a clean multiple of 4, you must erase exactly the leftover of 1.
- Which number carries leftover 1? Checking: 16 leaves 0, 17 leaves 1, 18 leaves 2, 19 leaves 3, 15 leaves 3. Only 17 has leftover 1, so erase 17.
- Why this transfers: for any divisibility question, work with remainders, not the big sums — a number's remainder is all that affects divisibility. Sanity check: 85 − 17 = 68 = 4 × 17. ✓
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