MathDB
5|(n + 6)^2+(n + 7)^2+ ... +(n + 10)^2 - [(n + 1)^2 + (n + 2)^2 +...+ (n + 5)]

Source: Netherlands - Dutch NMO 1965 p2

January 31, 2023
number theoryPerfect Squaredividesdivisible

Problem Statement

Prove that S1=(n+1)2+(n+2)2+...+(n+5)2S_1 = (n + 1)^2 + (n + 2)^2 +...+ (n + 5)^2 is divisible by 55 for every nn. Prove that for no nn: ∑ℓ=15(n+ℓ)2\sum_{\ell=1}^5 (n+\ell)^2 is a perfect square. Let S2=(n+6)2+(n+7)2+...+(n+10)2S_2=(n + 6)^2 + (n + 7)^2 + ... + (n + 10)^2. Prove that S1⋅S2S_1 \cdot S_2 is divisible by 150150.
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