Problem 14 · 2017 Math Kangaroo
Hard
Algebra & Patterns
Number Theory
substitutionperfect-square
The sum of the squares of three consecutive positive whole numbers is 770. What is the biggest of these numbers?
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Answer: C — 17
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Hint 1 of 2
Call the middle number n and write the three squares around it.
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Hint 2 of 2
The cross terms cancel, leaving a tidy equation in n.
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Approach: centre the three numbers on n so the linear terms cancel
- Let the numbers be n−1, n, n+1. Then (n−1)² + n² + (n+1)² = 3n² + 2.
- Set 3n² + 2 = 770, so 3n² = 768 and n² = 256, giving n = 16.
- The biggest number is n + 1 = 17, choice C.
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