Math worksheet for practicing finding the greatest common factor using prime factorization.

Problem Description:

The task is to find the Greatest Common Factor (GCF) of pairs of numbers using the method of Prime Factorization. The steps involve:
1. Finding the prime factorization of each number.
2. Identifying the common prime factors.
3. Multiplying the shared prime factors to get the GCF.

We are given four pairs of numbers:
- Pair 1: 45 and 66
- Pair 2: 24 and 56
- Pair 3: 42 and 70
- Pair 4: 72 and 96

Solution:

#### Step 1: Prime Factorization of Each Number

##### Pair 1: 45 and 66
- Prime Factorization of 45:
- $ 45 \div 3 = 15 $
- $ 15 \div 3 = 5 $
- $ 5 \div 5 = 1 $
- So, $ 45 = 3 \times 3 \times 5 = 3^2 \times 5 $

- Prime Factorization of 66:
- $ 66 \div 2 = 33 $
- $ 33 \div 3 = 11 $
- $ 11 \div 11 = 1 $
- So, $ 66 = 2 \times 3 \times 11 $

- Common Prime Factors:
- The only common prime factor is $ 3 $.

- GCF:
- Multiply the common prime factors: $ 3 $
- Therefore, $ \text{GCF}(45, 66) = 3 $

##### Pair 2: 24 and 56
- Prime Factorization of 24:
- $ 24 \div 2 = 12 $
- $ 12 \div 2 = 6 $
- $ 6 \div 2 = 3 $
- $ 3 \div 3 = 1 $
- So, $ 24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3 $

- Prime Factorization of 56:
- $ 56 \div 2 = 28 $
- $ 28 \div 2 = 14 $
- $ 14 \div 2 = 7 $
- $ 7 \div 7 = 1 $
- So, $ 56 = 2 \times 2 \times 2 \times 7 = 2^3 \times 7 $

- Common Prime Factors:
- The common prime factors are $ 2 \times 2 \times 2 = 2^3 $.

- GCF:
- Multiply the common prime factors: $ 2^3 = 8 $
- Therefore, $ \text{GCF}(24, 56) = 8 $

##### Pair 3: 42 and 70
- Prime Factorization of 42:
- $ 42 \div 2 = 21 $
- $ 21 \div 3 = 7 $
- $ 7 \div 7 = 1 $
- So, $ 42 = 2 \times 3 \times 7 $

- Prime Factorization of 70:
- $ 70 \div 2 = 35 $
- $ 35 \div 5 = 7 $
- $ 7 \div 7 = 1 $
- So, $ 70 = 2 \times 5 \times 7 $

- Common Prime Factors:
- The common prime factors are $ 2 \times 7 $.

- GCF:
- Multiply the common prime factors: $ 2 \times 7 = 14 $
- Therefore, $ \text{GCF}(42, 70) = 14 $

##### Pair 4: 72 and 96
- Prime Factorization of 72:
- $ 72 \div 2 = 36 $
- $ 36 \div 2 = 18 $
- $ 18 \div 2 = 9 $
- $ 9 \div 3 = 3 $
- $ 3 \div 3 = 1 $
- So, $ 72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2 $

- Prime Factorization of 96:
- $ 96 \div 2 = 48 $
- $ 48 \div 2 = 24 $
- $ 24 \div 2 = 12 $
- $ 12 \div 2 = 6 $
- $ 6 \div 2 = 3 $
- $ 3 \div 3 = 1 $
- So, $ 96 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 = 2^5 \times 3 $

- Common Prime Factors:
- The common prime factors are $ 2 \times 2 \times 2 \times 3 = 2^3 \times 3 $.

- GCF:
- Multiply the common prime factors: $ 2^3 \times 3 = 8 \times 3 = 24 $
- Therefore, $ \text{GCF}(72, 96) = 24 $

Final Answers:

\[
\boxed{3, 8, 14, 24}
\]