If n is a positive integer that is divisible by 6 but not divisible by 4, and n^2 has exac… — GRE General Test

If $n$ is a positive integer that is divisible by $6$ but not divisible by $4$ , and $n^2$ has exactly $15$ positive divisors, what is the value of $n$ ?

A
$6$
B
$12$
$18$ Answer
D
$24$
E
$36$

Answer

The value of

n

18

Because

n

is divisible by

6

, it must have prime factors

2

and

3

. Since

n

is not divisible by

4

, the exponent of

2

n

is exactly

1

, meaning

n^2

has

2^2

as a factor. The divisor count formula for

n^2

requires

(2+1)(2b+1) = 15

, which gives

2b+1 = 5

, so

b=2

. Thus,

n = 2^1 \cdot 3^2 = 18

Step-by-Step Solution

Analyze the prime factorization structure of

n

based on given divisibility conditions.

Since

n

is divisible by

6 = 2 \cdot 3

, its prime factorization must contain at least one factor of

2

and one factor of

3

. Since

n

is not divisible by

4 = 2^2

, the exponent of

2

in the prime factorization of

n

must be exactly

1

. Thus,

n = 2^1 \cdot 3^b \cdot k

, where

k

contains no factors of

2

3

Establishing the exponent of

2

narrows down the search space for prime factor exponents.

Express

n^2

in terms of its prime factors and write the formula for its number of positive divisors.

n^2 = 2^2 \cdot 3^{2b} \cdot k^2

. The number of positive divisors of

n^2

is given by

d(n^2) = (2 + 1)(2b + 1) \cdot d(k^2) = 3(2b + 1) \cdot d(k^2) = 15

The total number of positive divisors of a number

p_1^{a_1} p_2^{a_2} \dots

(a_1 + 1)(a_2 + 1) \dots

Solve for

b

and determine if

k

has any additional prime factors.

Dividing

15

3

gives

(2b + 1) \cdot d(k^2) = 5

. Since

5

is prime, we must have

d(k^2) = 1

(meaning

k = 1

) and

2b + 1 = 5

, which yields

2b = 4 \implies b = 2

Determining the exponent of

3

fixes the exact prime factorization of

n

Calculate

n

n = 2^1 \cdot 3^2 = 18

Multiplying the prime factors together yields the target integer.

Key Concept

Prime Factorization and Divisor Count Formula

Estimated Time:1m 30s

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