Problem 2 · 2001 AMC 8
Easy
Algebra & Patterns
factor-pairssum-constraint
I'm thinking of two whole numbers. Their product is 24 and their sum is 11. What is the larger number?
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Answer: D — 8.
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Hint 1 of 2
The product 24 is the strong clue — it only has a few factor pairs, so list those first and the sum filters them.
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Hint 2 of 2
"Two numbers with a known product and sum" is a classic setup; scanning factor pairs beats setting up algebra here.
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Approach: scan the factor pairs of 24
- Start from the product, because 24 has only four factor pairs: 1·24, 2·12, 3·8, 4·6.
- Now apply the sum filter — which pair adds to 11? Only 3 + 8 = 11.
- The larger of the two is 8. Whenever you know two numbers' product AND sum, list the factor pairs first; you'll meet the same idea later as factoring x² − 11x + 24.
Another way — guess-and-adjust from the sum:
- Split 11 and test: 5 + 6 → 30, 4 + 7 → 28, 3 + 8 → 24. ✓
- The pair 3 and 8 hits the product, so the larger is 8.
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