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?
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