Problem:
A positive integer has 60 divisors and has 80 divisors. What is the greatest integer such that divides ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let , where is the product of primes, none of which is 7 . Let be the number of divisors of . Then has divisors. Also , so has divisors. Thus
Hence . Note that meets the conditions of the problem.
The problems on this page are the property of the MAA's American Mathematics Competitions