Using prime factorisation, find the HCF and LCM of :
, (i) 8,9,25 (ii) 12,15,21 (iii) 17 ,23,29
(iv) 24,36,40 (v) 30,72,432 (vi) 21,28,36,45

Answer

To find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of the given sets of numbers using prime factorization, we will follow these steps for each part of the question.

### (i) For the numbers 8, 9, and 25:
1. **Prime Factorization**:
- 8 = 2 × 2 × 2 = \(2^3\)
- 9 = 3 × 3 = \(3^2\)
- 25 = 5 × 5 = \(5^2\)

2. **Finding HCF**:
- There are no common prime factors among 8, 9, and 25.
- Therefore, HCF = 1.

3. **Finding LCM**:
- LCM is found by taking the highest power of each prime factor present.
- LCM = \(2^3 \times 3^2 \times 5^2\)
- LCM = \(8 \times 9 \times 25 = 1800\).

### (ii) For the numbers 12, 15, and 21:
1. **Prime Factorization**:
- 12 = 2 × 2 × 3 = \(2^2 \times 3^1\)
- 15 = 3 × 5 = \(3^1 \times 5^1\)
- 21 = 3 × 7 = \(3^1 \times 7^1\)

2. **Finding HCF**:
- The common prime factor is 3.
- HCF = \(3^1 = 3\).

3. **Finding LCM**:
- LCM = \(2^2 \times 3^1 \times 5^1 \times 7^1\)
- LCM = \(4 \times 3 \times 5 \times 7 = 420\).

### (iii) For the numbers 17, 23, and 29:
1. **Prime Factorization**:
- 17 = \(17^1\)
- 23 = \(23^1\)
- 29 = \(29^1\)

2. **Finding HCF**:
- There are no common prime factors.
- Therefore, HCF = 1.

3. **Finding LCM**:
- LCM = \(17^1 \times 23^1 \times 29^1\)
- LCM = \(17 \times 23 \times 29 = 4039\).

### (iv) For the numbers 24, 36, and 40:
1. **Prime Factorization**:
- 24 = 2 × 2 × 2 × 3 = \(2^3 \times 3^1\)
- 36 = 2 × 2 × 3 × 3 = \(2^2 \times 3^2\)
- 40 = 2 × 2 × 2 × 5 = \(2^3 \times 5^1\)

2. **Finding HCF**:
- The common prime factor is \(2^2\).
- HCF = \(2^2 = 4\).

3. **Finding LCM**:
- LCM = \(2^3 \times 3^2 \times 5^1\)
- LCM = \(8 \times 9 \times 5 = 360\).

### (v) For the numbers 30, 72, and 432:
1. **Prime Factorization**:
- 30 = 2 × 3 × 5 = \(2^1 \times 3^1 \times 5^1\)
- 72 = 2 × 2 × 2 × 3 × 3 = \(2^3 \times 3^2\)
- 432 = 2 × 2 × 2 × 3 × 3 × 3 = \(2^4 \times 3^3\)

2. **Finding HCF**:
- The common prime factors are \(2^1\) and \(3^1\).
- HCF = \(2^1 \times 3^1 = 6\).

3. **Finding LCM**:
- LCM = \(2^4 \times 3^3 \times 5^1\)
- LCM = \(16 \times 27 \times 5 = 2160\).

### (vi) For the numbers 21, 28, 36, and 45:
1. **Prime Factorization**:
- 21 = 3 × 7 = \(3^1 \times 7^1\)
- 28 = 2 × 2 × 7 = \(2^2 \times 7^1\)
- 36 = 2 × 2 × 3 × 3 = \(2^2 \times 3^2\)
- 45 = 3 × 3 × 5 = \(3^2 \times 5^1\)

2. **Finding HCF**:
- The common prime factor is 3.
- HCF = \(3^1 = 3\).

3. **Finding LCM**:
- LCM = \(2^2 \times 3^2 \times 5^1 \times 7^1\)
- LCM = \(4 \times 9 \times 5 \times 7 = 1260\).

### Summary of Results:
- (i) HCF = 1, LCM = 1800
- (ii) HCF = 3, LCM = 420
- (iii) HCF = 1, LCM = 4039
- (iv) HCF = 4, LCM = 360
- (v) HCF = 6, LCM = 2160
- (vi) HCF = 3, LCM = 1260