Let’s solve each LCM problem step by step using prime factorization (the tree method shown in the worksheet).
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Problem 1: LCM of 50 and 75
Factor 50:
- 50 = 2 × 25 → 25 = 5 × 5 → so 50 = 2 × 5 × 5
Factor 75:
- 75 = 3 × 25 → 25 = 5 × 5 → so 75 = 3 × 5 × 5
Now, for LCM, take the highest power of each prime that appears:
- Primes involved: 2, 3, 5
- Highest powers: 2¹, 3¹, 5²
→ LCM = 2 × 3 × 5 × 5 = 6 × 25 =
150
Check:
Multiples of 50: 50, 100, 150...
Multiples of 75: 75, 150... → Yes, 150 is first common multiple.
✔ So:
50 = 2 × 5 × 5
75 = 3 × 5 × 5
LCM = 150
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Problem 2: LCM of 40 and 60
Factor 40:
- 40 = 2 × 20 → 20 = 2 × 10 → 10 = 2 × 5 → so 40 = 2 × 2 × 2 × 5 = 2³ × 5
Factor 60:
- 60 = 2 × 30 → 30 = 2 × 15 → 15 = 3 × 5 → so 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
Take highest powers:
- 2³, 3¹, 5¹ → LCM = 8 × 3 × 5 = 24 × 5 =
120
Check:
Multiples of 40: 40, 80, 120...
Multiples of 60: 60, 120... → Correct.
✔ So:
40 = 2 × 2 × 2 × 5
60 = 2 × 2 × 3 × 5
LCM = 120
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Problem 3: LCM of 24 and 36
Factor 24:
- 24 = 2 × 12 → 12 = 2 × 6 → 6 = 2 × 3 → so 24 = 2 × 2 × 2 × 3 = 2³ × 3
Factor 36:
- 36 = 2 × 18 → 18 = 2 × 9 → 9 = 3 × 3 → so 36 = 2 × 2 × 3 × 3 = 2² × 3²
Highest powers:
- 2³, 3² → LCM = 8 × 9 =
72
Check:
Multiples of 24: 24, 48, 72...
Multiples of 36: 36, 72... → Correct.
✔ So:
24 = 2 × 2 × 2 × 3
36 = 2 × 2 × 3 × 3
LCM = 72
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Problem 4: LCM of 24 and 56
We already have 24 = 2³ × 3
Factor 56:
- 56 = 2 × 28 → 28 = 2 × 14 → 14 = 2 × 7 → so 56 = 2 × 2 × 2 × 7 = 2³ × 7
Highest powers:
- 2³, 3¹, 7¹ → LCM = 8 × 3 × 7 = 24 × 7 =
168
Check:
Multiples of 24: 24, 48, 72, 96, 120, 144, 168...
Multiples of 56: 56, 112, 168... → Correct.
✔ So:
24 = 2 × 2 × 2 × 3
56 = 2 × 2 × 2 × 7
LCM = 168
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Final Answer:
Top-left:
50 = 2 × 5 × 5
75 = 3 × 5 × 5
LCM = 150
Top-right:
40 = 2 × 2 × 2 × 5
60 = 2 × 2 × 3 × 5
LCM = 120
Bottom-left:
24 = 2 × 2 × 2 × 3
36 = 2 × 2 × 3 × 3
LCM = 72
Bottom-right:
24 = 2 × 2 × 2 × 3
56 = 2 × 2 × 2 × 7
LCM = 168
Parent Tip: Review the logic above to help your child master the concept of lcm worksheets.
