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s(k) = 1 + z + z^2 + ...+ z^k divisible by n

Source: 1984 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p1

November 21, 2022
number theorydivisibledivides

Problem Statement

The natural numbers nn and zz are relatively prime and greater than 11. For k=0,1,2,...,n−1k = 0, 1, 2,..., n - 1 let s(k)=1+z+z2+...+zk.s(k) = 1 + z + z^2 + ...+ z^k. Prove that: a) At least one of the numbers s(k)s(k) is divisible by nn. b) If nn and z−1z - 1 are also coprime, then already one of the numbers s(k)s(k) with k=0,1,2,...,n−2k = 0,1, 2,..., n- 2 is divisible by nn.
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