AMC 12B 2022 (Problem 4)
For how many values of the constant will the polynomial have two distinct integer roots?
Since the roots are integers and
and must be integer divisors of .
and must be integer divisors of .
By Toolkit 33 — Number of Positive Divisors, the number of positive divisors of is
Therefore, has integer divisors: positive and negative.
By Hints 1, 2, and 3, assume without loss of generality that , since swapping the roots does not affect
The possible pairs and corresponding values of are
The pairs and are excluded because the roots must be distinct.
Thus, there are possible values of .
The possible pairs and corresponding values of are
The pairs and are excluded because the roots must be distinct.
Thus, there are possible values of .
(B)