Question

The prime factorizations of 54 and 72 are shown below.

Prime factorization of 54: 2, 3, 3, 3
Prime factorization of 72: 2, 2, 2, 3, 3

Using the prime factorizations, what is the greatest common factor of 54 and 72?
2 ´ 3
2 ´ 3 ´ 3
2 ´ 2 ´ 3 ´ 3
2 ´ 2 ´ 2 ´ 3 ´ 3 ´ 3

Answer 1

To find the greatest common factor (GCF) of 54 and 72 using their prime factorizations, we first list out the prime factorizations:

- 54: \(2^1 \times 3^3\)
- 72: \(2^3 \times 3^2\)

Next, we identify the lowest power of each common prime factor:

- For 2, the lowest power is \(2^1\) (from 54).
- For 3, the lowest power is \(3^2\) (from 72).

Now, we multiply these common prime factors together:

\[
\text{GCF} = 2^1 \times 3^2 = 2 \times 9 = 18
\]

So, looking at the provided options:

- \(2 \times 3\) = 6
- \(2 \times 3 \times 3\) = 18 (this is the correct answer!)
- \(2 \times 2 \times 3 \times 3\) = 36
- \(2 \times 2 \times 2 \times 3 \times 3 \times 3\) = 108

Thus, the greatest common factor of 54 and 72 is \(2 \times 3 \times 3\) or \(18\).