The prime factorizations of 54 and 72 are shown below.

Prime factorization of 54: [tex]\( 2 \times 3 \times 3 \times 3 \)[/tex]

Prime factorization of 72: [tex]\( 2 \times 2 \times 2 \times 3 \times 3 \)[/tex]

Using the prime factorizations, what is the greatest common factor of 54 and 72?

A. [tex]\( 2 \times 3 \)[/tex]

B. [tex]\( 2 \times 3 \times 3 \)[/tex]

C. [tex]\( 2 \times 2 \times 3 \times 3 \)[/tex]

D. [tex]\( 2 \times 2 \times 2 \times 3 \times 3 \times 3 \)[/tex]



Answer :

To find the greatest common factor (GCF) of 54 and 72, we start with their prime factorizations:

- The prime factorization of 54 is [tex]\(2 \times 3 \times 3 \times 3\)[/tex].
- The prime factorization of 72 is [tex]\(2 \times 2 \times 2 \times 3 \times 3\)[/tex].

Now, we determine which prime factors are common to both numbers and use the lowest powers of these common primes.

1. Common Factors:
- Both 54 and 72 have the prime factor [tex]\(2\)[/tex].
- Both 54 and 72 have the prime factor [tex]\(3\)[/tex].

2. Count the minimum occurrences of each common factor:
- For the prime factor [tex]\(2\)[/tex], the minimum count is 1 (since 54 has one 2, and 72 has three 2's).
- For the prime factor [tex]\(3\)[/tex], the minimum count is 2 (since 54 has three 3's, and 72 has two 3's).

To find the GCF, we construct it using these minimum occurrences:

[tex]\[ \text{GCF} = 2^1 \times 3^2 \][/tex]

Calculating this gives:

[tex]\[ \text{GCF} = 2 \times 9 = 18 \][/tex]

Therefore, the greatest common factor of 54 and 72 is [tex]\(18\)[/tex]. The correct option, if we list as multiples, is:

[tex]\[ 2 \times 3 \times 3 = 18 \][/tex]

So, the GCF of 54 and 72 is [tex]\(\boxed{18}\)[/tex].