A positive integer n is not divisible by 3. If n has exactly 15 positive divisors and 2n h… — GMAT

A positive integer $n$ is not divisible by $3$ . If $n$ has exactly $15$ positive divisors and $2n$ has exactly $20$ positive divisors, how many positive divisors does $3n$ have?

Answer: 30

Answer

The number of positive divisors of an integer with prime factorization

p_1^{e_1} p_2^{e_2} \dots

(e_1 + 1)(e_2 + 1) \dots

. Given

d(n) = 15

n

can be

p^{14}

p^4 q^2

. Given

d(2n) = 20

, 2 must be a prime factor of

n

with exponent 2, making

n = 2^2 p^4

. Since

n

is not divisible by 3,

p \neq 3

. Multiplying

n

by 3 introduces

3^1

into the prime factorization, giving

3n = 2^2 \cdot 3^1 \cdot p^4

. The total number of positive divisors is

(2+1)(1+1)(4+1) = 30

Step-by-Step Solution

Analyze the number of positive divisors of

n

n

is either

p^{14}

p^4 q^2

for distinct primes

p

and

q

The number of positive divisors of

n = p_1^{e_1} p_2^{e_2} \dots

is given by

(e_1 + 1)(e_2 + 1) \dots

. Since

15 = 15 \times 1 = 5 \times 3

, the exponent structure must be 14 or 4 and 2.

Analyze the number of positive divisors of

2n

n = 2^2 p^4

, where

p

is a prime other than 2 and 3.

n = p^4 q^2

, multiplying by 2 increases the exponent of 2 by 1. If

q = 2

, then

n = 2^2 p^4

and

2n = 2^3 p^4

, yielding

(3 + 1)(4 + 1) = 20

divisors. Any other prime structure yields a different count.

Determine the prime factorization of

3n

and count its positive divisors.

The number of divisors of

3n

is 30.

Since

n

is not divisible by 3,

p \neq 3

. Thus

3n = 2^2 \cdot 3^1 \cdot p^4

, which has

(2 + 1)(1 + 1)(4 + 1) = 30

positive divisors.

Key Concept

Divisor Count Formula and Prime Factorization

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