A certain positive integer has exactly 20 positive divisors.

a) What is the smallest number of primes that could divide the integer?

b) What is the largest number of primes that could divide the integer?

Badada Mar 22, 2019

#1**0 **

a) We could have two primes that divide the integer

Suppose that the number can be prime-factored into a^m * b^n

So

(m + 1) ( n + 1) = 20

And m, n could be (in some order)

3, 4 or

1, 9

b ) We could have three primes that divide the integer

Suppose that the integer can be prime-factored into a^k * b *m * c ^n

So

(k + 1) (m + 1) ( n + 1) = 20

And k, m, n could be (in some order)

1, 1, 4

CPhill Mar 22, 2019