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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🔹 Concepts:
-
GCF: The largest number that divides both numbers evenly.
-
LCM: The smallest number that is a multiple of both numbers.
- Formula:
$$
\text{LCM}(a, b) = \frac{a \times b}{\text{GCF}(a, b)}
$$
We'll use
prime factorization or
Euclidean algorithm to find GCF, then use the formula above for LCM.
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1. 60, 66
Prime factorization:
- 60 = 2² × 3 × 5
- 66 = 2 × 3 × 11
GCF: Common factors → 2 × 3 =
6
LCM: Take highest powers:
- 2² × 3 × 5 × 11 = 4 × 3 × 5 × 11 =
660
✔ GCF =
6, LCM =
660
---
2. 44, 14
Prime factorization:
- 44 = 2² × 11
- 14 = 2 × 7
GCF: Common factor → 2
LCM: 2² × 7 × 11 = 4 × 7 × 11 =
308
✔ GCF =
2, LCM =
308
---
3. 7, 56
- 7 is prime.
- 56 = 7 × 8 = 7 × 2³
GCF: 7 (since 7 divides 56)
LCM: Since 56 is a multiple of 7 → LCM =
56
✔ GCF =
7, LCM =
56
---
4. 20, 22
Prime factorization:
- 20 = 2² × 5
- 22 = 2 × 11
GCF: 2
LCM: 2² × 5 × 11 = 4 × 5 × 11 =
220
✔ GCF =
2, LCM =
220
---
5. 13, 31
Both are
prime numbers.
GCF: 1 (no common factors other than 1)
LCM: 13 × 31 =
403
✔ GCF =
1, LCM =
403
---
6. 8, 42
Prime factorization:
- 8 = 2³
- 42 = 2 × 3 × 7
GCF: 2
LCM: 2³ × 3 × 7 = 8 × 3 × 7 =
168
✔ GCF =
2, LCM =
168
---
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 Answers:
| Problem | 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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Let me know if you'd like a printable version or explanation with diagrams!
Parent Tip: Review the logic above to help your child master the concept of greatest common factor and least common multiple worksheet.
