Problem:
The positive integers x and y are the two smallest positive integers for which the product of 360 and x is a square and the product of 360 and y is a cube. What is the sum of x and y?
Answer Choices:
A. 80
B. 85
C. 115
D. 165
E. 610
Solution:
Factor 360 into 2â‹…2â‹…2â‹…3â‹…3â‹…5. First increase the number of each factor as little as possible to form a square: 2â‹…2â‹…2â‹…2â‹…3â‹…3â‹…5â‹…5=(2â‹…2â‹…2â‹…3â‹…3â‹…5)(2â‹…5)= (360)(10), so x is 10. Then increase the number of each factor as little as possible to form a cube: 2â‹…2â‹…2â‹…3â‹…3â‹…3â‹…5â‹…5â‹…5=(2â‹…2â‹…2â‹…3â‹…3â‹…5)(3â‹…5â‹…5)=(360)(75), so y is 75. The sum of x and y is 10+75=85.
Answer: B​.
The problems on this page are the property of the MAA's American Mathematics Competitions