MathDB

11

Part of 1977 IMO Longlists

Problems(1)

n must divide one of z_i [ILL 1977]

Source:

1/11/2011
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.
pigeonhole principlemodular arithmeticnumber theory proposednumber theory