Problem:
Prove that if 9 | a^2 + b^2 + c^2 then 9 divides at least one of a^2-b^2, b^2-c^2 or c^2 - a^2 .
Solution:
Proof by contradiction. Suppose none of them is divisible by 9. Then a^2, b^2, c^2 \pmod 9 are all different. A square number is 0, 1, -2, 4 \pmod 9 . But a^2 + b^2 + c^2 \equiv 1-2+4, 0-2+4, 0+1+4, 0+1-2 \pmod 9 \not \equiv 0 \pmod 9. QED
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