FREE Greatest Common Factor Worksheets [PDFs] Brighterly.com - Free Printable

To solve the problem of finding the Greatest Common Factor (GCF) and Least Common Multiple (LCM) for each pair of integers, we will use the following methods:

Step 1: Find the GCF

The GCF of two numbers is the largest number that divides both of them without leaving a remainder. We can find it using the prime factorization method or by listing the factors.

Step 2: Find the LCM

The LCM of two numbers is the smallest number that is a multiple of both. We can find it using the formula:
\[
\text{LCM}(a, b) = \frac{|a \times b|}{\text{GCF}(a, b)}
\]

Let's solve each pair step by step:

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#### Pair 1: 60 and 66

1. Find the GCF:
- Prime factorization of 60: \( 60 = 2^2 \times 3 \times 5 \)
- Prime factorization of 66: \( 66 = 2 \times 3 \times 11 \)
- Common factors: \( 2 \) and \( 3 \)
- GCF: \( 2 \times 3 = 6 \)

2. Find the LCM:
- Using the formula: \( \text{LCM}(60, 66) = \frac{60 \times 66}{\text{GCF}(60, 66)} = \frac{60 \times 66}{6} = \frac{3960}{6} = 660 \)

Result:
- GCF: \( 6 \)
- LCM: \( 660 \)

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#### Pair 2: 44 and 14

1. Find the GCF:
- Prime factorization of 44: \( 44 = 2^2 \times 11 \)
- Prime factorization of 14: \( 14 = 2 \times 7 \)
- Common factors: \( 2 \)
- GCF: \( 2 \)

2. Find the LCM:
- Using the formula: \( \text{LCM}(44, 14) = \frac{44 \times 14}{\text{GCF}(44, 14)} = \frac{44 \times 14}{2} = \frac{616}{2} = 308 \)

Result:
- GCF: \( 2 \)
- LCM: \( 308 \)

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#### Pair 3: 7 and 56

1. Find the GCF:
- Prime factorization of 7: \( 7 = 7 \)
- Prime factorization of 56: \( 56 = 2^3 \times 7 \)
- Common factors: \( 7 \)
- GCF: \( 7 \)

2. Find the LCM:
- Using the formula: \( \text{LCM}(7, 56) = \frac{7 \times 56}{\text{GCF}(7, 56)} = \frac{7 \times 56}{7} = \frac{392}{7} = 56 \)

Result:
- GCF: \( 7 \)
- LCM: \( 56 \)

---

#### Pair 4: 20 and 22

1. Find the GCF:
- Prime factorization of 20: \( 20 = 2^2 \times 5 \)
- Prime factorization of 22: \( 22 = 2 \times 11 \)
- Common factors: \( 2 \)
- GCF: \( 2 \)

2. Find the LCM:
- Using the formula: \( \text{LCM}(20, 22) = \frac{20 \times 22}{\text{GCF}(20, 22)} = \frac{20 \times 22}{2} = \frac{440}{2} = 220 \)

Result:
- GCF: \( 2 \)
- LCM: \( 220 \)

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#### Pair 5: 13 and 31

1. Find the GCF:
- Prime factorization of 13: \( 13 = 13 \)
- Prime factorization of 31: \( 31 = 31 \)
- Common factors: None (both are prime and distinct)
- GCF: \( 1 \)

2. Find the LCM:
- Using the formula: \( \text{LCM}(13, 31) = \frac{13 \times 31}{\text{GCF}(13, 31)} = \frac{13 \times 31}{1} = 403 \)

Result:
- GCF: \( 1 \)
- LCM: \( 403 \)

---

#### Pair 6: 8 and 42

1. Find the GCF:
- Prime factorization of 8: \( 8 = 2^3 \)
- Prime factorization of 42: \( 42 = 2 \times 3 \times 7 \)
- Common factors: \( 2 \)
- GCF: \( 2 \)

2. Find the LCM:
- Using the formula: \( \text{LCM}(8, 42) = \frac{8 \times 42}{\text{GCF}(8, 42)} = \frac{8 \times 42}{2} = \frac{336}{2} = 168 \)

Result:
- GCF: \( 2 \)
- LCM: \( 168 \)

---

#### Pair 7: 16 and 60

1. Find the GCF:
- Prime factorization of 16: \( 16 = 2^4 \)
- Prime factorization of 60: \( 60 = 2^2 \times 3 \times 5 \)
- Common factors: \( 2^2 = 4 \)
- GCF: \( 4 \)

2. Find the LCM:
- Using the formula: \( \text{LCM}(16, 60) = \frac{16 \times 60}{\text{GCF}(16, 60)} = \frac{16 \times 60}{4} = \frac{960}{4} = 240 \)

Result:
- GCF: \( 4 \)
- LCM: \( 240 \)

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Final Answers:

| Pair | GCF | LCM |
|------------|-----|------|
| 1. 60, 66 | 6 | 660 |
| 2. 44, 14 | 2 | 308 |
| 3. 7, 56 | 7 | 56 |
| 4. 20, 22 | 2 | 220 |
| 5. 13, 31 | 1 | 403 |
| 6. 8, 42 | 2 | 168 |
| 7. 16, 60 | 4 | 240 |

\boxed{
\begin{array}{|c|c|c|}
\hline
\text{Pair} & \text{GCF} & \text{LCM} \\
\hline
1. 60, 66 & 6 & 660 \\
2. 44, 14 & 2 & 308 \\
3. 7, 56 & 7 & 56 \\
4. 20, 22 & 2 & 220 \\
5. 13, 31 & 1 & 403 \\
6. 8, 42 & 2 & 168 \\
7. 16, 60 & 4 & 240 \\
\hline
\end{array}
}