Problem:
Let be the number of ordered pairs of integers , with and , such that the polynomial can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when is divided by .
Solution:
The factoring condition is equivalent to the discriminant being equal to for some integer . Because , the equation shows that the existence of such a is equivalent to with . Thus the number of ordered pairs is
The requested remainder is .
The problems on this page are the property of the MAA's American Mathematics Competitions