Question

What is the greatest common factor of 72, 90, and 54?

Answer 1

To find the greatest common factor (GCF) of 72, 90, and 54, we can start by determining the prime factorization of each of the numbers.

1. **Prime factorization of 72**:
- \( 72 = 2^3 \times 3^2 \)

2. **Prime factorization of 90**:
- \( 90 = 2^1 \times 3^2 \times 5^1 \)

3. **Prime factorization of 54**:
- \( 54 = 2^1 \times 3^3 \)

Next, we identify the common prime factors and select the smallest power of each common factor from all three numbers.

- For the prime factor \(2\):
- The minimum power in the factorizations is \(2^1\) (since \(72\) has \(2^3\), \(90\) has \(2^1\), and \(54\) has \(2^1\)).

- For the prime factor \(3\):
- The minimum power is \(3^2\) (since \(72\) has \(3^2\), \(90\) has \(3^2\), and \(54\) has \(3^3\)).

Now we can find the GCF by multiplying the smallest powers of the common prime factors:

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

Therefore, the greatest common factor of 72, 90, and 54 is \(\boxed{18}\).