Problem:
Let A be the set of positive integers that have no prime factors other than 2,3, or 5. The infinite sum
11+21+31+41+51+61+81+91+101+121+151+161+181+201+⋯
of the reciprocals of all the elements of A can be expressed as nm, where m and n are relatively prime positive integers. What is m+n?
Answer Choices:
A. 16
B. 17
C. 19
D. 23
E. 36
Solution:
Elements of set A are of the form 2i⋅3j⋅5k for nonnegative integers i,j, and k. Note that the product
(1+21+221+⋯)(1+31+321+⋯)(1+51+521+⋯)
will produce the desired sum. By the formula for infinite geometric series, this product evaluates to
1−211⋅1−311⋅1−511=2⋅23⋅45=415
The requested sum is 15+4=19.
The problems on this page are the property of the MAA's American Mathematics Competitions