Question
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Answer
- **a)** LCM of 8 and 12 is **24**
- **b)** LCM of 16 and 20 is **80**
- **c)** LCM of 21 and 18 is **126**
Solution
The **Least Common Multiple (LCM)** of two numbers is the smallest positive integer that is a multiple of both numbers. Let's find the LCM for each pair:
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### a) LCM of 8 and 12
1. **Prime Factorization:**
- **8** = \(2^3\)
- **12** = \(2^2 \times 3\)
2. **Determine LCM:**
- Take the highest power of each prime factor.
- LCM = \(2^3 \times 3 = 8 \times 3 = 24\)
**LCM(8, 12) = 24**
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### b) LCM of 16 and 20
1. **Prime Factorization:**
- **16** = \(2^4\)
- **20** = \(2^2 \times 5\)
2. **Determine LCM:**
- Take the highest power of each prime factor.
- LCM = \(2^4 \times 5 = 16 \times 5 = 80\)
**LCM(16, 20) = 80**
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### c) LCM of 21 and 18
1. **Prime Factorization:**
- **21** = \(3 \times 7\)
- **18** = \(2 \times 3^2\)
2. **Determine LCM:**
- Take the highest power of each prime factor.
- LCM = \(2 \times 3^2 \times 7 = 2 \times 9 \times 7 = 126\)
**LCM(21, 18) = 126**
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### Summary
- **a)** LCM of 8 and 12 is **24**
- **b)** LCM of 16 and 20 is **80**
- **c)** LCM of 21 and 18 is **126**
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Simplify this solution Bonus Knowledge
To find the Least Common Multiple (L.C.M), we can use the prime factorization approach or the relationship between the Greatest Common Divisor (G.C.D) and L.C.M. For 8 and 12, the prime factors are 2³ (for 8) and 2² × 3¹ (for 12). The L.C.M is calculated by taking the highest power of each prime: L.C.M = 2³ × 3¹ = 24. For 16 and 20, the prime factors are 2⁴ (for 16) and 2² × 5¹ (for 20). The L.C.M here is 2⁴ × 5¹ = 80. For 21 and 18, the factors are 3² (for 18) and 7¹ × 3¹ (for 21). Therefore, the L.C.M = 2⁰ × 3² × 7¹ = 126. So, the L.C.M of each set is: a) 24 b) 80 c) 126
