Let's solve each of the Least Common Multiple (LCM) problems step by step.
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What is LCM?
The
Least Common Multiple (LCM) of a set of numbers is the smallest number that is a multiple of all the numbers in the set.
We can find the LCM using:
1.
Prime factorization method – break down each number into prime factors, then take the highest power of each prime that appears.
2.
Listing multiples – list multiples of each number until you find the smallest common one (works for small numbers).
We'll use
prime factorization for efficiency.
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Problem 1: LCM(20, 10, 30)
- 20 = 2² × 5
- 10 = 2 × 5
- 30 = 2 × 3 × 5
Take the highest powers of all primes:
- 2², 3¹, 5¹ → 4 × 3 × 5 =
60
✔ LCM = 60
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Problem 2: LCM(15, 25, 30)
- 15 = 3 × 5
- 25 = 5²
- 30 = 2 × 3 × 5
Highest powers:
- 2¹, 3¹, 5² → 2 × 3 × 25 =
150
✔ LCM = 150
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Problem 3: LCM(12, 36, 20)
- 12 = 2² × 3
- 36 = 2² × 3²
- 20 = 2² × 5
Highest powers:
- 2², 3², 5¹ → 4 × 9 × 5 =
180
✔ LCM = 180
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Problem 4: LCM(25, 40, 30)
- 25 = 5²
- 40 = 2³ × 5
- 30 = 2 × 3 × 5
Highest powers:
- 2³, 3¹, 5² → 8 × 3 × 25 =
600
✔ LCM = 600
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Problem 5: LCM(25, 20, 15)
- 25 = 5²
- 20 = 2² × 5
- 15 = 3 × 5
Highest powers:
- 2², 3¹, 5² → 4 × 3 × 25 =
300
✔ LCM = 300
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Problem 6: LCM(36, 12, 45)
- 36 = 2² × 3²
- 12 = 2² × 3
- 45 = 3² × 5
Highest powers:
- 2², 3², 5¹ → 4 × 9 × 5 =
180
✔ LCM = 180
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Problem 7: LCM(24, 9, 15)
- 24 = 2³ × 3
- 9 = 3²
- 15 = 3 × 5
Highest powers:
- 2³, 3², 5¹ → 8 × 9 × 5 =
360
✔ LCM = 360
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Problem 8: LCM(20, 10, 15)
- 20 = 2² × 5
- 10 = 2 × 5
- 15 = 3 × 5
Highest powers:
- 2², 3¹, 5¹ → 4 × 3 × 5 =
60
✔ LCM = 60
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Problem 9: LCM(24, 36, 12)
- 24 = 2³ × 3
- 36 = 2² × 3²
- 12 = 2² × 3
Highest powers:
- 2³, 3² → 8 × 9 =
72
✔ LCM = 72
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Problem 10: LCM(45, 25, 50)
- 45 = 3² × 5
- 25 = 5²
- 50 = 2 × 5²
Highest powers:
- 2¹, 3², 5² → 2 × 9 × 25 =
450
✔ LCM = 450
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✔ Final Answers:
| Problem | LCM |
|--------|-----|
| 1. LCM(20, 10, 30) |
60 |
| 2. LCM(15, 25, 30) |
150 |
| 3. LCM(12, 36, 20) |
180 |
| 4. LCM(25, 40, 30) |
600 |
| 5. LCM(25, 20, 15) |
300 |
| 6. LCM(36, 12, 45) |
180 |
| 7. LCM(24, 9, 15) |
360 |
| 8. LCM(20, 10, 15) |
60 |
| 9. LCM(24, 36, 12) |
72 |
|10. LCM(45, 25, 50) |
450 |
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Let me know if you'd like a printable version or explanation using another method (like listing multiples)!
Parent Tip: Review the logic above to help your child master the concept of least common multiples worksheet 5th grade.
