Relatively Prime: Two integers are said to be relatively prime if there
is no integer greater than one that divides them both.

(a): Factorization of 15 is 15 = 3 * 5 and the factorization of 25 is

25 = 5 × 5

Since 5 is the common factor in factorization of both the numbers, therefore they are not prime numbers.

(b): Factorization of 29 is 29 = 1 × 29 and the factorization of 58 is 58 = 2 × 29

Since 29 is the common factor in factorization of both the numbers, therefore they are not prime numbers.

(c): Factorization of 40 is 40 = 1 × 2 × 2 × 2 × 5

and the factorization of 63 is 63 = 1 × 3 × 3 × 7

Since in the factorization of both the numbers, there is no common number than one, thus 40 and 63 are prime numbers.

(d): Factorization of 261 is 261 = 3 × 3 × 29

and the factorization of 513 is 513 = 3 × 3 × 3 × 19 Since 3 is the common factor in factorization of both the numbers, therefore they are not prime numbers. Thus, the pair 261 and 513 are not relatively prime.

(a): Factorization of 15 is 15 = 3 * 5 and the factorization of 25 is

25 = 5 × 5

Since 5 is the common factor in factorization of both the numbers, therefore they are not prime numbers.

(b): Factorization of 29 is 29 = 1 × 29 and the factorization of 58 is 58 = 2 × 29

Since 29 is the common factor in factorization of both the numbers, therefore they are not prime numbers.

(c): Factorization of 40 is 40 = 1 × 2 × 2 × 2 × 5

and the factorization of 63 is 63 = 1 × 3 × 3 × 7

Since in the factorization of both the numbers, there is no common number than one, thus 40 and 63 are prime numbers.

(d): Factorization of 261 is 261 = 3 × 3 × 29

and the factorization of 513 is 513 = 3 × 3 × 3 × 19 Since 3 is the common factor in factorization of both the numbers, therefore they are not prime numbers. Thus, the pair 261 and 513 are not relatively prime.