Which of the following is a pair of co-prime numbers?

Options

A. 8, 15
B. 3, 18
C. 5, 35
D. 6, 39

Answer

To determine which of the given pairs of numbers are co-prime, we need to follow these steps:

### Step 1: Understand Co-prime Numbers
Co-prime numbers are two numbers that do not have any common factors other than 1. This means that their greatest common divisor (GCD) is 1.

### Step 2: Factor Each Pair of Numbers
We will factor each pair of numbers to find their prime factors.

1. **Pair 1: 8 and 15**
- Factors of 8: \( 2 \times 2 \times 2 \) (or \( 2^3 \))
- Factors of 15: \( 3 \times 5 \)

2. **Pair 2: 18 and 35**
- Factors of 18: \( 2 \times 3 \times 3 \) (or \( 2 \times 3^2 \))
- Factors of 35: \( 5 \times 7 \)

3. **Pair 3: 6 and 39**
- Factors of 6: \( 2 \times 3 \)
- Factors of 39: \( 3 \times 13 \)

4. **Pair 4: 5 and 5**
- Factors of 5: \( 5 \) (itself, since 5 is a prime number)

### Step 3: Check for Common Factors
Now we will check each pair for common factors:

1. **For 8 and 15:**
- Factors: \( 2, 2, 2 \) (from 8) and \( 3, 5 \) (from 15)
- No common factors. **Co-prime.**

2. **For 18 and 35:**
- Factors: \( 2, 3, 3 \) (from 18) and \( 5, 7 \) (from 35)
- No common factors. **Co-prime.**

3. **For 6 and 39:**
- Factors: \( 2, 3 \) (from 6) and \( 3, 13 \) (from 39)
- Common factor: \( 3 \). **Not co-prime.**

4. **For 5 and 5:**
- Factors: \( 5 \) (from both)
- Common factor: \( 5 \). **Not co-prime.**

### Conclusion
The pairs of co-prime numbers from the given options are:
- 8 and 15
- 18 and 35