Question

What is the greatest common factor of 72 and 9?

Answer 1

To find the greatest common factor (GCF) of 72 and 9, we can start by determining the prime factorizations of both numbers.

**Prime Factorization of 72:**
1. Divide by 2: \( 72 \div 2 = 36 \)
2. Divide by 2: \( 36 \div 2 = 18 \)
3. Divide by 2: \( 18 \div 2 = 9 \)
4. Divide by 3: \( 9 \div 3 = 3 \)
5. Divide by 3: \( 3 \div 3 = 1 \)

So, the prime factorization of 72 is:
\[
72 = 2^3 \times 3^2
\]

**Prime Factorization of 9:**
1. Divide by 3: \( 9 \div 3 = 3 \)
2. Divide by 3: \( 3 \div 3 = 1 \)

So, the prime factorization of 9 is:
\[
9 = 3^2
\]

**Finding the GCF:**
Now, we can identify the common prime factors and take the lowest powers:

- For the prime factor 3, the minimum exponent in both factorizations is \( 2 \) (from both 72 and 9).

Thus, the GCF can be calculated as:
\[
GCF = 3^2 = 9
\]

Therefore, the greatest common factor of 72 and 9 is \( \boxed{9} \).