Problem 26 · 2011 Math Kangaroo
Stretch
Number Theory
divisibility
Determine the sum of all positive whole numbers x less than 100 for which \(x^{2}-81\) is a multiple of 100.
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Answer: A — 200
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Hint 1 of 2
x² − 81 is a multiple of 100 means x² ≡ 81 (mod 100).
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Hint 2 of 2
Solve modulo 4 and modulo 25 separately, then combine.
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Approach: solve the quadratic congruence mod 4 and mod 25
- x must be odd (mod 4) and x ≡ ±9 (mod 25).
- Combining gives x = 9, 41, 59, 91 below 100.
- Their sum is 9+41+59+91 = 200.
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