A positive integer N has no prime factors other than 2 and 3. If N is a multiple of 12 and… — GMAT

A positive integer $N$ has no prime factors other than $2$ and $3$ . If $N$ is a multiple of $12$ and has exactly $18$ positive divisors, what is the sum of all possible values of $N$ ?

Answer: 2028

Answer

2028

The correct numerical answer is 2028, obtained by determining all exponent combinations of 2 and 3 that yield 18 total factors while guaranteeing divisibility by 12.

Step-by-Step Solution

Set up the prime factorization of

N

with constraints.

N = 2^a \cdot 3^b

with

a \ge 2

and

b \ge 1

Since the only prime factors are

2

and

3

, and

12 = 2^2 \cdot 3^1

divides

N

, the exponents must satisfy

a \ge 2

and

b \ge 1

Apply the total number of divisors formula.

(a+1)(b+1) = 18

The number of positive divisors of

p_1^{e_1} p_2^{e_2}

(e_1 + 1)(e_2 + 1)

Identify all valid integer solution pairs for

(a+1, b+1)

Valid pairs are

(9,2)

(6,3)

, and

(3,6)

, corresponding to

(a,b) = (8,1), (5,2), (2,5)

We require

a+1 \ge 3

and

b+1 \ge 2

. Pairs

(18,1)

and

(2,9)

violate these lower bounds.

Compute the corresponding values of

N

and find their sum.

768 + 288 + 972 = 2028

Evaluating

2^8 \cdot 3^1 = 768

2^5 \cdot 3^2 = 288

, and

2^2 \cdot 3^5 = 972

yields a total sum of

2028

Key Concept

Divisor Count Formula & Multiplicativity Constraints

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