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Step-by-step solution for: least common multiple | PDF
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Show Answer Key & Explanations
Step-by-step solution for: least common multiple | PDF
You've uploaded an image of a math worksheet titled “Least Common Multiple”, which asks students to find the LCM (Least Common Multiple) for five pairs of numbers by listing their multiples and identifying the smallest common one.
Let’s go through each problem step-by-step and explain the solution clearly.
---
Multiples of 4:
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ...
*(These are all numbers you get by multiplying 4 × 1, 4 × 2, 4 × 3, etc.)*
Multiples of 12:
12, 24, 36, 48, 60, 72, 84, 96, 108, 120, ...
✔ The first number that appears in both lists is 12.
➡️ LCM(4, 12) = 12
> 💡 *Note: Since 12 is a multiple of 4 (12 ÷ 4 = 3), the LCM is just the larger number — 12.*
---
Multiples of 18:
18, 36, 54, 72, 90, 108, 126, 144, 162, 180, ...
Multiples of 24:
24, 48, 72, 96, 120, 144, 168, 192, 216, 240, ...
✔ The first common multiple is 72.
➡️ LCM(18, 24) = 72
> 💡 *You can also use prime factorization:*
> - 18 = 2 × 3²
> - 24 = 2³ × 3
> - LCM = highest powers: 2³ × 3² = 8 × 9 = 72 ✔
---
Multiples of 3:
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...
Multiples of 9:
9, 18, 27, 36, 45, 54, 63, 72, 81, 90, ...
✔ First common multiple is 9.
➡️ LCM(3, 9) = 9
> 💡 *Since 9 is a multiple of 3, LCM is 9.*
---
Multiples of 21:
21, 42, 63, 84, 105, 126, 147, 168, 189, 210, ...
Multiples of 28:
28, 56, 84, 112, 140, 168, 196, 224, 252, 280, ...
✔ First common multiple is 84.
➡️ LCM(21, 28) = 84
> 💡 *Prime factorization check:*
> - 21 = 3 × 7
> - 28 = 2² × 7
> - LCM = 2² × 3 × 7 = 4 × 3 × 7 = 84 ✔
---
Multiples of 45:
45, 90, 135, 180, 225, 270, 315, 360, 405, 450, ...
Multiples of 27:
27, 54, 81, 108, 135, 162, 189, 216, 243, 270, ...
✔ First common multiple is 135.
➡️ LCM(45, 27) = 135
> 💡 *Prime factorization:*
> - 45 = 3² × 5
> - 27 = 3³
> - LCM = 3³ × 5 = 27 × 5 = 135 ✔
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## ✔ Final Answer Key:
| Problem | Numbers | LCM |
|---------|-------------|------|
| 1 | 4, 12 | 12 |
| 2 | 18, 24 | 72 |
| 3 | 3, 9 | 9 |
| 4 | 21, 28 | 84 |
| 5 | 45, 27 | 135 |
---
## 📝 Teaching Tip:
The method used here — listing multiples — is great for beginners because it’s visual and concrete. For larger numbers, students will eventually learn more efficient methods like:
- Prime Factorization (as shown above)
- Division Method / Ladder Method
- Using the formula:
LCM(a, b) = (a × b) ÷ GCF(a, b)
But for now, listing multiples helps build number sense and understanding of what “least common multiple” really means!
Let me know if you’d like to see any of these solved using another method! 😊
Let’s go through each problem step-by-step and explain the solution clearly.
---
🔢 Problem 1: LCM(4, 12)
Multiples of 4:
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ...
*(These are all numbers you get by multiplying 4 × 1, 4 × 2, 4 × 3, etc.)*
Multiples of 12:
12, 24, 36, 48, 60, 72, 84, 96, 108, 120, ...
✔ The first number that appears in both lists is 12.
➡️ LCM(4, 12) = 12
> 💡 *Note: Since 12 is a multiple of 4 (12 ÷ 4 = 3), the LCM is just the larger number — 12.*
---
🔢 Problem 2: LCM(18, 24)
Multiples of 18:
18, 36, 54, 72, 90, 108, 126, 144, 162, 180, ...
Multiples of 24:
24, 48, 72, 96, 120, 144, 168, 192, 216, 240, ...
✔ The first common multiple is 72.
➡️ LCM(18, 24) = 72
> 💡 *You can also use prime factorization:*
> - 18 = 2 × 3²
> - 24 = 2³ × 3
> - LCM = highest powers: 2³ × 3² = 8 × 9 = 72 ✔
---
🔢 Problem 3: LCM(3, 9)
Multiples of 3:
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...
Multiples of 9:
9, 18, 27, 36, 45, 54, 63, 72, 81, 90, ...
✔ First common multiple is 9.
➡️ LCM(3, 9) = 9
> 💡 *Since 9 is a multiple of 3, LCM is 9.*
---
🔢 Problem 4: LCM(21, 28)
Multiples of 21:
21, 42, 63, 84, 105, 126, 147, 168, 189, 210, ...
Multiples of 28:
28, 56, 84, 112, 140, 168, 196, 224, 252, 280, ...
✔ First common multiple is 84.
➡️ LCM(21, 28) = 84
> 💡 *Prime factorization check:*
> - 21 = 3 × 7
> - 28 = 2² × 7
> - LCM = 2² × 3 × 7 = 4 × 3 × 7 = 84 ✔
---
🔢 Problem 5: LCM(45, 27)
Multiples of 45:
45, 90, 135, 180, 225, 270, 315, 360, 405, 450, ...
Multiples of 27:
27, 54, 81, 108, 135, 162, 189, 216, 243, 270, ...
✔ First common multiple is 135.
➡️ LCM(45, 27) = 135
> 💡 *Prime factorization:*
> - 45 = 3² × 5
> - 27 = 3³
> - LCM = 3³ × 5 = 27 × 5 = 135 ✔
---
## ✔ Final Answer Key:
| Problem | Numbers | LCM |
|---------|-------------|------|
| 1 | 4, 12 | 12 |
| 2 | 18, 24 | 72 |
| 3 | 3, 9 | 9 |
| 4 | 21, 28 | 84 |
| 5 | 45, 27 | 135 |
---
## 📝 Teaching Tip:
The method used here — listing multiples — is great for beginners because it’s visual and concrete. For larger numbers, students will eventually learn more efficient methods like:
- Prime Factorization (as shown above)
- Division Method / Ladder Method
- Using the formula:
LCM(a, b) = (a × b) ÷ GCF(a, b)
But for now, listing multiples helps build number sense and understanding of what “least common multiple” really means!
Let me know if you’d like to see any of these solved using another method! 😊
Parent Tip: Review the logic above to help your child master the concept of lcm worksheet with answers.
