Find the L.C.M. of the given numbers by prime factorisation method:
(i) 28, 98 `" "` (ii) 36, 40, 120 `" "` (iii) 108, 135, 162 `" "` (iv) 24, 28, 196.

Answer

To find the L.C.M. (Least Common Multiple) of the given numbers using the prime factorization method, we will follow these steps for each set of numbers.

### (i) L.C.M. of 28 and 98

**Step 1: Prime Factorization**
- **28**:
- 28 = 2 × 14
- 14 = 2 × 7
- So, the prime factorization of 28 = 2² × 7¹
- **98**:
- 98 = 2 × 49
- 49 = 7 × 7
- So, the prime factorization of 98 = 2¹ × 7²

**Step 2: Identify the highest power of each prime factor**
- For 2: max(2², 2¹) = 2²
- For 7: max(7¹, 7²) = 7²

**Step 3: Calculate the L.C.M.**
- L.C.M. = 2² × 7² = 4 × 49 = 196

### (ii) L.C.M. of 36, 40, and 120

**Step 1: Prime Factorization**
- **36**:
- 36 = 6 × 6
- 6 = 2 × 3
- So, the prime factorization of 36 = 2² × 3²
- **40**:
- 40 = 8 × 5
- 8 = 2³
- So, the prime factorization of 40 = 2³ × 5¹
- **120**:
- 120 = 12 × 10
- 12 = 2² × 3
- 10 = 2 × 5
- So, the prime factorization of 120 = 2³ × 3¹ × 5¹

**Step 2: Identify the highest power of each prime factor**
- For 2: max(2², 2³, 2³) = 2³
- For 3: max(3², 3¹) = 3²
- For 5: max(5¹, 5¹) = 5¹

**Step 3: Calculate the L.C.M.**
- L.C.M. = 2³ × 3² × 5¹ = 8 × 9 × 5 = 360

### (iii) L.C.M. of 108, 135, and 162

**Step 1: Prime Factorization**
- **108**:
- 108 = 36 × 3
- 36 = 6 × 6
- 6 = 2 × 3
- So, the prime factorization of 108 = 2² × 3³
- **135**:
- 135 = 27 × 5
- 27 = 3³
- So, the prime factorization of 135 = 3³ × 5¹
- **162**:
- 162 = 54 × 3
- 54 = 18 × 3
- 18 = 2 × 9
- 9 = 3²
- So, the prime factorization of 162 = 2¹ × 3⁴

**Step 2: Identify the highest power of each prime factor**
- For 2: max(2², 2¹) = 2²
- For 3: max(3³, 3³, 3⁴) = 3⁴
- For 5: max(5¹) = 5¹

**Step 3: Calculate the L.C.M.**
- L.C.M. = 2² × 3⁴ × 5¹ = 4 × 81 × 5 = 1620

### (iv) L.C.M. of 24, 28, and 196

**Step 1: Prime Factorization**
- **24**:
- 24 = 6 × 4
- 6 = 2 × 3
- 4 = 2²
- So, the prime factorization of 24 = 2³ × 3¹
- **28**:
- As calculated earlier, 28 = 2² × 7¹
- **196**:
- 196 = 14 × 14
- 14 = 2 × 7
- So, the prime factorization of 196 = 2² × 7²

**Step 2: Identify the highest power of each prime factor**
- For 2: max(2³, 2², 2²) = 2³
- For 3: max(3¹) = 3¹
- For 7: max(7¹, 7²) = 7²

**Step 3: Calculate the L.C.M.**
- L.C.M. = 2³ × 3¹ × 7² = 8 × 3 × 49 = 1176

### Final Answers:
1. L.C.M. of 28 and 98 is **196**.
2. L.C.M. of 36, 40, and 120 is **360**.
3. L.C.M. of 108, 135, and 162 is **1620**.
4. L.C.M. of 24, 28, and 196 is **1176**.