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If n is a good number, what is the minimum number of divisors that n^2 has?

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We call a positive integer a "good number", if the product of all its divisors equals its cube.

For example, 12 is a good number, because the divisors of 12 are 1, 2, 3, 4, 6, 12, and 1*2*3*4*6*12=1728=12^3.

If n is a good number, what is the minimum number of divisors that n^2 has?

posted Oct 29, 2015 by Sandeep Bedi

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See the following table

Number    Number of positive divisors
1           1
2*2         3
3*3*3       4
4*4*4*4     9 

As n increases, the number of positive divisors of nxnxnxn.....n (n times) increases. Is it true or false?

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