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].
