Which of the following are co-primes ?

Options

A. 8 , 12
B. 9 , 10
C. 6, 8
D. 15, 18

Answer

To determine which of the given pairs of numbers are co-prime, we need to find the highest common factor (HCF) of each pair. Co-prime numbers are defined as numbers that have no common factors other than 1, meaning their HCF is 1.

Let's analyze each pair step by step:

### Step 1: Analyze the pair (8, 12)
1. **Find the factors of 8**:
- 8 = 2 × 2 × 2
- Factors of 8: 1, 2, 4, 8

2. **Find the factors of 12**:
- 12 = 2 × 2 × 3
- Factors of 12: 1, 2, 3, 4, 6, 12

3. **Find the HCF**:
- Common factors of 8 and 12: 1, 2, 4
- HCF = 4

4. **Conclusion**: Since the HCF is not 1, **8 and 12 are not co-prime**.

### Step 2: Analyze the pair (9, 10)
1. **Find the factors of 9**:
- 9 = 3 × 3
- Factors of 9: 1, 3, 9

2. **Find the factors of 10**:
- 10 = 2 × 5
- Factors of 10: 1, 2, 5, 10

3. **Find the HCF**:
- Common factors of 9 and 10: 1
- HCF = 1

4. **Conclusion**: Since the HCF is 1, **9 and 10 are co-prime**.

### Step 3: Analyze the pair (6, 8)
1. **Find the factors of 6**:
- 6 = 2 × 3
- Factors of 6: 1, 2, 3, 6

2. **Find the factors of 8**:
- (Already calculated above)
- Factors of 8: 1, 2, 4, 8

3. **Find the HCF**:
- Common factors of 6 and 8: 1, 2
- HCF = 2

4. **Conclusion**: Since the HCF is not 1, **6 and 8 are not co-prime**.

### Step 4: Analyze the pair (15, 18)
1. **Find the factors of 15**:
- 15 = 3 × 5
- Factors of 15: 1, 3, 5, 15

2. **Find the factors of 18**:
- 18 = 2 × 3 × 3
- Factors of 18: 1, 2, 3, 6, 9, 18

3. **Find the HCF**:
- Common factors of 15 and 18: 1, 3
- HCF = 3

4. **Conclusion**: Since the HCF is not 1, **15 and 18 are not co-prime**.

### Final Conclusion:
- The only pair that are co-prime numbers is **9 and 10**.