RO-TST 1988 Problem 10

Let $p > 2$ be a prime number. Find the least positive number $a$ which can be represented as $$a = (X - 1)f(X) + (X^{p - 1} + X^{p - 2} + \cdots + X + 1)g(X),$$ where $f(X)$ and $g(X)$ are integer polynomials.

Mircea Becheanu.