Which of the following are co-primes ?

Options

A. `91` and `72`
B. `34` and `51`
C. `21` and `36`
D. `15` and `20`

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. If the HCF is 1, then the numbers are co-prime.

### Step-by-Step Solution:

1. **Identify the pairs of numbers**: The question provides several pairs of numbers. For this solution, we will check the first pair: 91 and 72.

2. **Find the prime factorization of each number**:
- For **91**:
- 91 can be factored into prime numbers as follows:
- 91 = 7 × 13
- For **72**:
- 72 can be factored into prime numbers as follows:
- 72 = 2 × 2 × 2 × 3 × 3 (or 2^3 × 3^2)

3. **List the prime factors**:
- Prime factors of 91: {7, 13}
- Prime factors of 72: {2, 3}

4. **Determine the common factors**:
- The common factors between the two sets of prime factors are checked.
- Since there are no common factors (the sets {7, 13} and {2, 3} share no numbers), we conclude that the only common factor is 1.

5. **Calculate the HCF**:
- The HCF of 91 and 72 is 1.

6. **Conclusion**:
- Since the HCF is 1, we conclude that **91 and 72 are co-prime**.

7. **Check other pairs**:
- For the second pair (34 and 51):
- 34 = 2 × 17
- 51 = 3 × 17
- HCF = 17 (not co-prime)
- For the third pair (21 and 36):
- 21 = 3 × 7
- 36 = 3 × 12
- HCF = 3 (not co-prime)
- For the fourth pair (15 and 20):
- 15 = 5 × 3
- 20 = 5 × 4
- HCF = 5 (not co-prime)

### Final Answer:
- The only co-prime pair from the options is **91 and 72**.