1. Let P(x) be a polynomial whose coefficients are positive integers. If P(n) divides P(P(n) -2015) for every natural number n, prove that P(-2015) = 0.

P(P(n)-2015) -P(-2015) will be divisible by (P(n)-2015) – (-2015) = P(n).

 

But we are given that P(n) divides P(P(n)-2015) and hence P(n) divides P(-2015) for any n

 

Since the coefficients are all natural numbers, P(n) becomes arbitrarily large as n becomes large. Hence P(-2015) is an integer that can be made as small in magnitude as we please, which means P(-2015) = 0