A positive integer K has exactly 12 positive factors. The greatest common divisor of K and… — ACT

A positive integer $K$ has exactly $12$ positive factors. The greatest common divisor of $K$ and $84$ is $6$ , and the least common multiple of $K$ and $24$ is $360$ . What is the value of $K$ ?

A
60
B
72
90Answer
D
120
E
180

Answer

The prime factorizations are

84 = 2^2 \times 3 \times 7

6 = 2 \times 3

24 = 2^3 \times 3

, and

360 = 2^3 \times 3^2 \times 5

. The condition

\gcd(K, 84) = 6

implies the power of 2 in

K

is exactly 1, and the power of 3 is at least 1. The condition

\text{lcm}(K, 24) = 360

implies the power of 3 in

K

is exactly 2, and the power of 5 is exactly 1. Thus,

K

must be

2^1 \times 3^2 \times 5^1 = 90

. Testing

90

, it has

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

positive factors, which perfectly satisfies all conditions.

Step-by-Step Solution

Find the prime factorization of the given numbers in the problem.

84 = 2^2 \times 3 \times 7

6 = 2 \times 3

24 = 2^3 \times 3

, and

360 = 2^3 \times 3^2 \times 5

Expressing these numbers in terms of their prime components allows us to determine the constraints on the prime factorization of

K

Analyze the greatest common divisor constraint

\gcd(K, 84) = 6

The prime 2 must have an exponent of exactly 1 in the factorization of

K

(since

84

has

2^2

and the greatest common divisor has only

2^1

). The prime 3 must have an exponent of at least 1 in

K

. The prime 7 cannot be a factor of

K

The greatest common divisor selects the minimum exponent for each shared prime factor.

Analyze the least common multiple constraint

\text{lcm}(K, 24) = 360

The prime 3 must have an exponent of exactly 2 in

K

(since

24

has

3^1

and the least common multiple has

3^2

). The prime 5 must have an exponent of exactly 1 in

K

(since

24

has

5^0

and the least common multiple has

5^1

K

cannot contain any prime factors other than 2, 3, and 5.

The least common multiple selects the maximum exponent for each prime factor present in either number.

Combine the exponent constraints to determine

K

and verify its factor count.

K = 2^1 \times 3^2 \times 5^1 = 90

. The number of positive factors of

90

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

This unique configuration matches all given greatest common divisor and least common multiple prime power requirements and satisfies the factor count condition.

Key Concept

Prime Factorization, Greatest Common Divisor, Least Common Multiple, and Number of Factors

Alternative Method

Instead of analyzing prime factor exponents, check the given multiple-choice options. Test each option by checking if it has exactly 12 factors, a greatest common divisor of 6 with 84, and a least common multiple of 360 with 24. Only the value 90 meets all three criteria.

Estimated Time:2m 30s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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