A positive integer n has only two distinct prime factors, 2 and 3. If n is a multiple of 1… — GMAT

A positive integer $n$ has only two distinct prime factors, $2$ and $3$ . If $n$ is a multiple of $12$ , is not divisible by $8$ , and has exactly $12$ positive divisors, what is the value of $n$ ?

Answer: 108

Answer

The value of

n

is 108.

Because

n

has only

2

and

3

as prime factors, its prime factorization is

n = 2^a \times 3^b

. Divisibility by

12 = 2^2 \times 3^1

requires

a \ge 2

, and non-divisibility by

8 = 2^3

requires

a < 3

, which forces

a = 2

. Using the formula for total positive divisors,

(2+1)(b+1) = 12

, which simplifies to

3(b+1) = 12

and gives

b = 3

. Calculating

n = 2^2 \times 3^3

yields

108

Step-by-Step Solution

Set up the prime factorization of

n

n = 2^a \times 3^b

where

a \ge 1

and

b \ge 1

The problem states that

2

and

3

are the only distinct prime factors of

n

Determine the exact value of exponent

a

a = 2

n

is divisible by

12 = 2^2 \times 3^1

(so

a \ge 2

) but not by

8 = 2^3

(so

a < 3

Determine the exact value of exponent

b

b = 3

The total number of positive divisors is

(a+1)(b+1) = (2+1)(b+1) = 3(b+1) = 12

, which solves to

b = 3

Calculate the value of

n

n = 108

n = 2^2 \times 3^3 = 4 \times 27 = 108

Key Concept

Determining integer values using prime factorization and the number of divisors formula

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic