Problem:
For , let , where there are zeros between the and the . Let be the number of factors of in the prime factorization of . What is the maximum value of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Note that . For , the first term is not divisible by , so . For , the first term is divisible by , but the second term is not, so . For , and because the second factor is even, . In fact the second factor is a sum of cubes so
The factor is divisible by but not , and the second factor is odd, so contributes one more factor of . Hence the maximum value for is .
The problems on this page are the property of the MAA's American Mathematics Competitions