To determine which pair of numbers is coprime, we need to find if their greatest common divisor (GCD) is 1. Let’s analyze each pair:
- a) 13 and 14: The GCD of 13 and 14 is 1 since 13 is a prime number and does not divide 14. Thus, 13 and 14 are coprime.
- b) 8 and 20: Both 8 and 20 share a GCD of 4 (since 8 = 2 × 4 and 20 = 5 × 4). Therefore, they are not coprime.
- c) 31 and 59: Both 31 and 59 are prime numbers, and they do not share any divisors other than 1, making their GCD also 1. Thus, they are coprime.
- d) 34 and 85: The GCD of 34 and 85 is 17 (since 34 = 2 × 17 and 85 = 5 × 17). Hence, they are not coprime.
In conclusion, the coprime pairs from the options provided are:
- a) 13 and 14
- c) 31 and 59
Both pairs have a GCD of 1, confirming they are coprime.