Question

What is the greatest common factor (GCF) of 54 and 45?
A.
6
B.
90
C.
9
D.
270

Answer 1

To find the greatest common factor (GCF) of 54 and 45, we can start by finding the prime factorizations of both numbers.

1. **Prime factorization of 54**:
- 54 = 2 × 27
- 27 = 3 × 9
- 9 = 3 × 3
- Therefore, the prime factorization of 54 is:
\[
54 = 2^1 × 3^3
\]

2. **Prime factorization of 45**:
- 45 = 5 × 9
- 9 = 3 × 3
- Therefore, the prime factorization of 45 is:
\[
45 = 3^2 × 5^1
\]

3. **Finding the GCF**:
- Now we compare the prime factors. The common prime factor is 3.
- In 54, the power of 3 is 3 (from \( 3^3 \)).
- In 45, the power of 3 is 2 (from \( 3^2 \)).
- We take the lowest power of the common prime factor:
\[
\text{GCF} = 3^{\min(3, 2)} = 3^2 = 9
\]

Thus, the greatest common factor (GCF) of 54 and 45 is **9**.

The answer is **C. 9**.