2005 AMC 10A Problems/Problem 21
For how many positive integers does evenly divide from ?
So the problem asks us for how many positive integers is an integer.
is an integer when is a factor of .
The factors of are , , , , , and .
So the possible values of are , , , , , and .
But isn't a positive integer, so only , , , , and are possible values of .
Therefore the number of possible values of is
The sum of the first positive integers is . If this is to divide , then there exists a positive integer such that:
Therefore, and are divisors of . There are divisors of , which are . The divisors which multiply to can be assigned to and in either order. However, when is assigned to , then , which is not possible, because must be positive. Therefore, we have values of