Which of the following numbers are co-prime? 

(a) 18 and 35            

(b) 15 and 37                 

(c) 30 and 415            

(d) 17 and 68            

(e) 216 and 215            

(f) 81 and 16

To determine which pairs of numbers are co-prime, we need to check if their greatest common divisor (GCD) is 1. If the GCD is 1, the numbers are co-prime. Here’s a brief explanation for each pair:

(a) 18 and 35: The GCD of 18 and 35 is 1, so they are co-prime.

Explanation: 18 (2 x 3²) and 35 (5 x 7) have no common prime factors.

(b) 15 and 37: The GCD of 15 and 37 is 1, so they are co-prime.

Explanation: 15 (3 x 5) and 37 (prime number) have no common factors.

(c) 30 and 415: The GCD of 30 and 415 is 5, so they are not co-prime.

Explanation: 30 (2 x 3 x 5) and 415 (5 x 83) share the common factor 5.

(d) 17 and 68: The GCD of 17 and 68 is 17, so they are not co-prime.

Explanation: 17 is a prime factor of 68 (17 x 4), so they are not co-prime.

(e) 216 and 215: The GCD of 216 and 215 is 1, so they are co-prime.

Explanation: 216 (2³ x 3³) and 215 (5 x 43) have no common factors.

(f) 81 and 16: The GCD of 81 and 16 is 1, so they are co-prime.

Explanation: 81 (3⁴) and 16 (2⁴) have no common prime factors.

Final Answer

• Co-prime pairs: (a) 18 and 35, (b) 15 and 37, (e) 216 and 215, (f) 81 and 16.
• Not co-prime pairs: (c) 30 and 415, (d) 17 and 68.