Numbers and Sets | Part IA, 2003

(i) Prove by induction or otherwise that for every n1n \geqslant 1,

r=1nr3=(r=1nr)2\sum_{r=1}^{n} r^{3}=\left(\sum_{r=1}^{n} r\right)^{2}

(ii) Show that the sum of the first nn positive cubes is divisible by 4 if and only if n0n \equiv 0 or 3(mod4)3(\bmod 4).

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