SOLUTION: A positive integer n has 60 divisors and 7n has 80 divisors. What is the greatest integer k such that 7^k divides n?

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Question 618060: A positive integer n has 60 divisors and 7n has 80 divisors. What is the greatest integer k such that 7^k divides n?
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
Let



where are the prime divisors of n. Then,

.

To count the number of factors of an integer, we take the k_i's, add 1 to each, and multiply. The number of factors of n is

The number of factors of 7n is

If we divide the first equation by the second equation, the products will cancel out, so we are left with



Therefore, , solving for k_1 we obtain . The answer is 2.