Problem:
For how many positive integers n does 1+2+⋯+n evenly divide 6n?
Answer Choices:
A. 3
B. 5
C. 7
D. 9
E. 11
Solution:
Because
1+2+⋯+n=2n(n+1)
1+2+⋯+n divides the positive integer 6n if and only if
n(n+1)/26n=n+112 is an integer.
There are 5 such positive values of n, namely, 1,2,3,5, and 11.
Answer: B.
The problems on this page are the property of the MAA's American Mathematics Competitions