Problem:
Let N=695+5⋅694+10⋅693+10⋅692+5⋅69+1. How many positive integers are factors of N?
Answer Choices:
A. 3
B. 5
C. 69
D. 125
E. 216
Solution:
By the Binomial Theorem, N=(69+1)5=(2⋅5⋅7)5. Thus a positive integer d is a factor of N iff d=2p5q7∗, where p,q,r are each one of the 6 integers 0,1,2,3,4,5. Therefore there are 63=216 choices for d.