Two numbers are said to be co-prime if the only number that divides both of the numbers is 1.

(a) 18 and 35

Factors of 18 = 1, 2, 3, 6, 9, and 18

Factors of 35 = 1, 5, 7 and 35

Common factor = 1

Therefore, the given two numbers are co-prime.

(b) 15 and 37

Factors of 15 = 1, 3, 5, and 15

Factors of 35 = 1, and 37

Common factor = 1

Therefore, the given two numbers are co-prime.

(c) 30 and 415

Factors of 30 = 1, 2, 3, 5, 6, 10, 15, and 30

Factors of 415 = 1, 5, 83 and 415

Common factor = 1 and 5

These two numbers have a factor other than 1, therefore, the given two numbers are not co-prime.

(d) 17 and 68

Factors of 17 = 1, and 17

Factors of 68 = 1, 2, 4, 17, 34 and 68

Common factor = 1, and 17

These two numbers have a factor other than 1, therefore, the given two numbers are not co-prime.

(e) 216 and 215

Factors of 216 = 1, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108 and 216

Factors of 215 = 1, 5, 43 and 215

Common factor = 1

Therefore, the given two numbers are co-prime.

(f) 81 and 16

Factors of 81 = 1, 3, 9, 27 and 81

Factors of 35 = 1, 2, 4, 8 and 16

Common factor = 1

Therefore, the given two numbers are co-prime.