Make 5 pairs of co-prime numbers from the following numbers: 2, 6, 8 `|` 2, 24, 35

Options

A. 2 and 35, 6 and 35, 8 and 35, 12 and 35, 24 and 35
B. 2 and 35, 6 and 35, 8 and 35, 12 and 35, 24 and 6
C. 2 and 35, 6 and 35, 8 and 35, 12 and 35, 24 and 8
D. None of these

Answer

To find 5 pairs of co-prime numbers from the given set of numbers (2, 6, 8, 24, 35), we need to identify pairs of numbers that share no common factors other than 1. Here’s how we can do this step by step:

### Step 1: Understand Co-prime Numbers
Co-prime numbers are pairs of numbers that have no common factors other than 1. This means their highest common factor (HCF) is 1.

### Step 2: List the Given Numbers
The numbers we have are:
- 2
- 6
- 8
- 24
- 35

### Step 3: Find Factors of Each Number
- **Factors of 2:** 1, 2
- **Factors of 6:** 1, 2, 3, 6
- **Factors of 8:** 1, 2, 4, 8
- **Factors of 24:** 1, 2, 3, 4, 6, 8, 12, 24
- **Factors of 35:** 1, 5, 7, 35

### Step 4: Identify Co-prime Pairs
Now, we will check pairs of numbers to see if they are co-prime (i.e., their HCF is 1):

1. **Pair (2, 35)**:
- Common factors: 1 (HCF = 1) → Co-prime

2. **Pair (6, 35)**:
- Common factors: 1 (HCF = 1) → Co-prime

3. **Pair (8, 35)**:
- Common factors: 1 (HCF = 1) → Co-prime

4. **Pair (2, 6)**:
- Common factors: 1, 2 (HCF = 2) → Not co-prime

5. **Pair (2, 24)**:
- Common factors: 1, 2 (HCF = 2) → Not co-prime

6. **Pair (6, 8)**:
- Common factors: 1, 2 (HCF = 2) → Not co-prime

7. **Pair (6, 24)**:
- Common factors: 1, 2, 3, 6 (HCF = 6) → Not co-prime

8. **Pair (8, 24)**:
- Common factors: 1, 2, 4, 8 (HCF = 8) → Not co-prime

### Step 5: Continue Finding Co-prime Pairs
We already found three pairs: (2, 35), (6, 35), and (8, 35). We need two more pairs.

9. **Pair (24, 35)**:
- Common factors: 1 (HCF = 1) → Co-prime

10. **Pair (2, 24)**:
- Common factors: 1, 2 (HCF = 2) → Not co-prime

11. **Pair (6, 24)**:
- Common factors: 1, 2, 3, 6 (HCF = 6) → Not co-prime

12. **Pair (8, 24)**:
- Common factors: 1, 2, 4, 8 (HCF = 8) → Not co-prime

### Final Co-prime Pairs
After checking all combinations, we can finalize our pairs:
1. (2, 35)
2. (6, 35)
3. (8, 35)
4. (24, 35)
5. (2, 6) → Not co-prime, so we need to find another.

### Conclusion
The final pairs of co-prime numbers are:
1. (2, 35)
2. (6, 35)
3. (8, 35)
4. (24, 35)
5. (6, 24) → Not co-prime, so we can take (2, 8) as a co-prime pair.

### Final Answer
The five pairs of co-prime numbers are:
1. (2, 35)
2. (6, 35)
3. (8, 35)
4. (24, 35)
5. (2, 6) → Not co-prime, so we take (2, 8).