Which of the following pairs of numbers are co-prime?

(i) 12 and 35

(ii) 15 and 37

(iii) 27 and 32

(iv) 17 and 85

(v) 515 and 516

(vi) 215 and 415

(i) The factors of 12 are 1, 2, 3, 4, 6 and 12.

The factors of 35 are 1, 5, 7 and 35.

The only common factor is 1.

Hence, 12 and 35 are co-prime.

(ii) The factors of 15 are 1, 3, 5 and 15.

The factors of 37 are 1 and 37 (since 37 is prime).

The only common factor is 1.

Hence, 15 and 37 are co-prime.

(iii) The factors of 27 are 1, 3, 9 and 27.

The factors of 32 are 1, 2, 4, 8, 16 and 32.

The only common factor is 1.

Hence, 27 and 32 are co-prime.

(iv) The factors of 17 are 1 and 17.

The factors of 85 are 1, 5, 17 and 85.

The common factors are 1 and 17.

Hence, 17 and 85 are not co-prime.

(v) 515 and 516 are two consecutive natural numbers.

Any two consecutive natural numbers are always co-prime.

Hence, 515 and 516 are co-prime.

(vi) The factors of 215 are 1, 5, 43 and 215.

The factors of 415 are 1, 5, 83 and 415.

The common factors are 1 and 5.

Hence, 215 and 415 are not co-prime.