Worksheet for calculating greatest common factor (GCF) and least common multiple (LCM) of integer pairs.

Let's solve each problem step by step to find the Greatest Common Factor (GCF) and Least Common Multiple (LCM) for each pair of integers.

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🔹 Definitions:

- GCF (Greatest Common Factor): The largest number that divides both numbers evenly.
- LCM (Least Common Multiple): The smallest number that is a multiple of both numbers.

We can use:
- Prime factorization or
- Euclidean algorithm for GCF
- Then use the formula:
$$
\text{LCM}(a, b) = \frac{a \times b}{\text{GCF}(a, b)}
$$

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✔ Problem 1: 60, 66

Step 1: Prime Factorization
- 60 = 2² × 3 × 5
- 66 = 2 × 3 × 11

GCF: Take common prime factors with lowest powers → 2 × 3 = 6

LCM: Take highest powers of all primes → 2² × 3 × 5 × 11 = 4 × 3 × 5 × 11 = 660

✔ GCF = 6, LCM = 660

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✔ Problem 2: 44, 14

Prime Factorization:
- 44 = 2² × 11
- 14 = 2 × 7

GCF: Common factor → 2 → 2

LCM: 2² × 7 × 11 = 4 × 7 × 11 = 308

✔ GCF = 2, LCM = 308

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✔ Problem 3: 7, 56

Note: 7 is prime.

- 7 = 7
- 56 = 7 × 8 = 7 × 2³

GCF: 7 → 7

LCM: Since 56 is divisible by 7, LCM = 56 → 56

✔ GCF = 7, LCM = 56

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✔ Problem 4: 20, 22

Prime Factorization:
- 20 = 2² × 5
- 22 = 2 × 11

GCF: 2 → 2

LCM: 2² × 5 × 11 = 4 × 5 × 11 = 220

✔ GCF = 2, LCM = 220

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✔ Problem 5: 13, 31

Both are prime numbers.

GCF: Only common factor is 1 → 1

LCM: 13 × 31 = 403

✔ GCF = 1, LCM = 403

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✔ Problem 6: 8, 42

Prime Factorization:
- 8 = 2³
- 42 = 2 × 3 × 7

GCF: 2 → 2

LCM: 2³ × 3 × 7 = 8 × 3 × 7 = 168

✔ GCF = 2, LCM = 168

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✔ Problem 7: 16, 60

Prime Factorization:
- 16 = 2⁴
- 60 = 2² × 3 × 5

GCF: 2² = 4

LCM: 2⁴ × 3 × 5 = 16 × 3 × 5 = 240

✔ GCF = 4, LCM = 240

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📊 Final Answer Table:

| 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 |

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✔ All problems solved!
Let me know if you'd like this as a printable answer sheet.