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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?

 Mar 22, 2019
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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

 

 

cool cool cool

 Mar 22, 2019

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