Problem: 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.
But it is given that for all n.
Hence for all n.
Note that P(-2015) is a fixed number, hence with finitely many divisors.
As is positive, by increasing n arbitrarily, we can increase the value of P(n) infinitely.
But infinitely many numbers cannot divide a finite number (P(-2015)) unless it is equal to 0.
There fore P(-2015) = 0.