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) (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  \prod_{i=2}^{m} (k_i + 1) = 60)
The number of factors of 7n is \prod_{i=2}^{m} (k_i + 1) = 80)
If we divide the first equation by the second equation, the products will cancel out, so we are left with
Therefore,  = 3(k_1 + 2))
, solving for k_1 we obtain 
. The answer is 2.
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