Printable math worksheet for practicing greatest common factor (GCF) problems, including prime factorization and factor listing methods.

It looks like you've uploaded a worksheet titled "Greatest Common Factor (GCF)", and it includes several problems asking students to find the GCF of pairs of numbers using different methods. Let's go through each part step by step, solve the problems, and explain the solutions.

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🔹 Part 1: Find the GCF using Prime Factorization

#### Example Given:
Find the GCF of 36 and 48.

Step 1: Prime factorization
- 36 = 2 × 2 × 3 × 3 = $2^2 \times 3^2$
- 48 = 2 × 2 × 2 × 2 × 3 = $2^4 \times 3^1$

Step 2: Identify common prime factors with lowest exponents
- Common primes: 2 and 3
- Lowest powers: $2^2$ and $3^1$

Step 3: Multiply
- GCF = $2^2 \times 3 = 4 \times 3 = 12$

✔ So, GCF of 36 and 48 is 12.

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🔹 Part 2: Use a Venn Diagram to Find GCF

This method involves listing prime factors in overlapping circles.

For example, for 36 and 48:

- Left circle (only in 36): $3^2$ (extra 3s)
- Right circle (only in 48): $2^2$ (extra 2s)
- Overlap (common): $2^2 \times 3 = 12$

So again, GCF = 12.

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🔹 Now Solve the Problems on the Worksheet

Let’s work through each one.

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#### ✔ Problem 1: GCF of 12 and 18

Step 1: Prime factorization
- 12 = $2^2 \times 3$
- 18 = $2 \times 3^2$

Common factors: $2^1 \times 3^1 = 2 \times 3 = 6$

✔ GCF = 6

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#### ✔ Problem 2: GCF of 30 and 75

Prime factorization:
- 30 = $2 \times 3 \times 5$
- 75 = $3 \times 5^2$

Common factors: $3 \times 5 = 15$

✔ GCF = 15

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#### ✔ Problem 3: GCF of 48 and 60

Prime factorization:
- 48 = $2^4 \times 3$
- 60 = $2^2 \times 3 \times 5$

Common factors: $2^2 \times 3 = 4 \times 3 = 12$

✔ GCF = 12

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#### ✔ Problem 4: GCF of 24 and 36

Prime factorization:
- 24 = $2^3 \times 3$
- 36 = $2^2 \times 3^2$

Common factors: $2^2 \times 3 = 4 \times 3 = 12$

✔ GCF = 12

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#### ✔ Problem 5: GCF of 40 and 60

Prime factorization:
- 40 = $2^3 \times 5$
- 60 = $2^2 \times 3 \times 5$

Common factors: $2^2 \times 5 = 4 \times 5 = 20$

✔ GCF = 20

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#### ✔ Problem 6: GCF of 70 and 100

Prime factorization:
- 70 = $2 \times 5 \times 7$
- 100 = $2^2 \times 5^2$

Common factors: $2 \times 5 = 10$

✔ GCF = 10

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🔹 Final Answers:

| Numbers | GCF |
|--------|-----|
| 12, 18 | 6 |
| 30, 75 | 15 |
| 48, 60 | 12 |
| 24, 36 | 12 |
| 40, 60 | 20 |
| 70, 100 | 10 |

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✔ Summary of Method:

To find the Greatest Common Factor (GCF) of two numbers:

1. Find the prime factorization of each number.
2. Identify common prime factors.
3. Multiply the lowest powers of the common primes.

Alternatively, use a Venn diagram or list of factors (but prime factorization is most efficient).

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Let me know if you'd like these answers formatted neatly for printing or if you want help with visualizing the Venn diagrams!