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.
We can use:
-
Prime factorization method
- 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, and different.
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 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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Parent Tip: Review the logic above to help your child master the concept of gcf and lcm worksheet pdf.
