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Turkey TST 1998 Problem 6, N is divisible by 13

Source: Turkey TST 1998 Problem 6

December 2, 2011
algebrapolynomialnumber theory proposednumber theorycombinatorial nullstellensatz

Problem Statement

Let f(x1,x2,...,xn)f(x_{1}, x_{2}, . . . , x_{n}) be a polynomial with integer coefficients of degree less than nn. Prove that if NN is the number of nn-tuples (x1,...,xn)(x_{1}, . . . , x_{n}) with 0≤xi<130 \leq x_{i} < 13 and f(x1,...,xn)=0(mod13)f(x_{1}, . . . , x_{n}) = 0 (mod 13), then NN is divisible by 13.
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