Problem:
How many positive integer divisors of are perfect squares or perfect cubes (or both)?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Because , a square divisor has the form where , and a cubic divisor has the form where . A number is both a square and a cube if and only if it is a sixth power, so it has the form where . Thus there are square divisors, cubic divisors, and divisors that are sixth powers. Therefore the number of divisors that are squares and/or cubes is .
The problems on this page are the property of the MAA's American Mathematics Competitions