If k is a positive integer that is divisible by both 8 and 9, and k has exactly 12 positiv… — GMAT

If $k$ is a positive integer that is divisible by both 8 and 9, and $k$ has exactly 12 positive integer divisors, what is the value of $k$ ?

The value 72 has the prime factorization

2^3 \cdot 3^2

. It is divisible by

8 = 2^3

and

9 = 3^2

, and its number of positive integer divisors is

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

, satisfying all conditions of the problem.

Express the divisibility conditions in terms of prime factors.

Since

k

is divisible by

8 = 2^3

and

9 = 3^2

, the prime factorization of

k

must be of the form

k = 2^a \cdot 3^b \cdot p_1^{c_1} \cdots

, where

a \geq 3

and

b \geq 2

Any multiple of 8 must contain at least three factors of 2, and any multiple of 9 must contain at least two factors of 3.

Write the formula for the total number of positive integer divisors.

The number of positive integer divisors is given by

(a+1)(b+1)(c_1+1)\cdots = 12

The total number of divisors of a number expressed in prime factorization is found by adding 1 to each exponent and multiplying the results.

Determine the values of the exponents

a

and

b

Since

a \geq 3

, we have

a+1 \geq 4

. Since

b \geq 2

, we have

b+1 \geq 3

. Therefore,

(a+1)(b+1) \geq 4 \times 3 = 12

Because the product

(a+1)(b+1)

is already at least 12, there can be no additional prime factors, so

a+1 = 4 \implies a = 3

and

b+1 = 3 \implies b = 2

Calculate the value of

k

k = 2^3 \cdot 3^2 = 8 \cdot 9 = 72

Evaluating

2^3 \cdot 3^2

gives the unique positive integer satisfying all given conditions.

Determining integer values using prime factorization constraints and total divisor count rules.

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