Question

Let n be a positive integer not divisible by 6. Suppose n has 6 positive divisors. The number of positive divisors of 9n is​

Answer 1

Given:

1. Let n be a positive integer not divisible by 6.

2. Suppose n has 6 positive divisors.

Asked: The number of positive divisors of 9n could be

A) 54

B) 36

C) 18

D) 12

E) More than one of the above

6 = 3*2 = 1*6

n can be of the form = p1^2*p2; where p1 & p2 are prime numbers

p1 & p2 can not be 2 & multiple of 3

or n can be of the form = p^5

Case 1: n is of the form = p1^2*p2

9n = 9*p1^2*p2 = 3^2*p1^2*p2

Either p1 or p2 can be 3.

If p1=3

9n = 3^4*p2

Number of positive divisors of 9n = 5*2 = 10

If p2 = 3

9n = 3^3*p1^2

Number of positive divisors of 9n = 4*3 = 12

If p1 & p2 are different than 3

9n = 9*p1^2*p2 = 3^2*p1^2*p2

Number of positive divisors of 9n = 3*3*2 = 18

Maximum number of divisors of 9n = 18

Case 1: n is of the form = p^5

9n = 3^2*p^5; where p is different than 3

Number of positive divisors = 3*6 = 18

But if p=3

9n = 3^7

Number of positive divisors = 8

Since 12 & 18 both are possible number of positive divisors

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