**Problem:Â **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.

**Discussion:Â **

Let

Then

Now note

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.

## Chatuspathi:

**Paper:**RMO 2015Â Mumbai**What is this topic:**Â Polynomial**What are some of the associated concepts:**Â Modular Arithmetic**Where can learn these topics:Â**Cheenta**Â**I.S.I. & C.M.I. course,Cheenta Math Olympiad Program, discuss these topics in the â€˜Polynomialâ€™ module.**Book Suggestions:Â**Polynomial by Barbeau

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