An integer k is defined as the Least Common Multiple (LCM) of 12 and 28. What is the Great… — Armed Services Vocational Aptitude Battery (ASVAB)

An integer $k$ is defined as the Least Common Multiple (LCM) of $12$ and $28$ . What is the Greatest Common Factor (GCF) of $k$ and $40$ ?

The Greatest Common Factor of

k

and

40

is 4.

The problem requires two distinct mathematical steps. First, calculate the Least Common Multiple of

12

and

28

. Using prime factorizations (

12 = 2^2 \times 3

and

28 = 2^2 \times 7

), the LCM is the product of the highest powers of all prime factors:

2^2 \times 3 \times 7 = 84

. Therefore,

k = 84

. The second step asks for the Greatest Common Factor of

k

(

84

) and

40

. The prime factorization of

40

2^3 \times 5

. The only shared prime factor between

84

and

40

2

, and the lowest power they share is

2^2

. Thus, the GCF is

4

Find the prime factorizations of 12 and 28.

12 = 2^2 \times 3

and

28 = 2^2 \times 7

Prime factorization is the most reliable method for finding the Least Common Multiple.

Calculate

k

, which is the LCM of 12 and 28.

k = 2^2 \times 3 \times 7 = 4 \times 21 = 84

The LCM is found by multiplying the highest power of every prime factor present in either number.

Find the prime factorization of 40.

40 = 2^3 \times 5

This is needed to compare against the prime factors of

k

(

84

) to find the Greatest Common Factor.

Calculate the GCF of 84 and 40.

Comparing

84

(

2^2 \times 3 \times 7

) and

40

(

2^3 \times 5

), the only shared prime base is 2. The lowest exponent for this base is 2. Thus, the GCF is

2^2 = 4

The GCF is the product of the lowest powers of all common prime factors shared by both integers.

Applying prime factorization to sequentially determine Least Common Multiples (LCM) and Greatest Common Factors (GCF).

Question