Let be the number of positive integer divisors of that leave a remainder of upon division by . Find the remainder when is divided by .
Note that . For a number to leave a remainder of 5 upon division by , it must leave a remainder of mod and mod .
Suppose our divisor was . Taking this modulo gives
so is even. Taking this modulo gives
so is odd.
Note are any integer between and (inclusive). Thus, any choice for results in choices of and , and can be chosen however. Thus, there are a total of
divisors that are modulo , for an answer of .
The problems and solutions on this page are the property of the MAA's American Mathematics Competitions