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n must divide one of z_i [ILL 1977]

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January 11, 2011
pigeonhole principlemodular arithmeticnumber theory proposednumber theory

Problem Statement

Let nn and zz be integers greater than 11 and (n,z)=1(n,z)=1. Prove: (a) At least one of the numbers zi=1+z+z2+⋯+zi, i=0,1,…,n−1,z_i=1+z+z^2+\cdots +z^i,\ i=0,1,\ldots ,n-1, is divisible by nn. (b) If (z−1,n)=1(z-1,n)=1, then at least one of the numbers ziz_i is divisible by nn.
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