To determine which of the given pairs of numbers are coprime, we first need to understand what it means for two numbers to be coprime. Two numbers are considered coprime if their greatest common divisor (GCD) is 1.
Let’s evaluate each pair of numbers step by step to find their GCD and see if they are coprime:
a) 51, 63
First, we need to determine the GCD of 51 and 63:
- The prime factorization of 51 is [tex]\(3 \times 17\)[/tex].
- The prime factorization of 63 is [tex]\(3^2 \times 7\)[/tex].
The common factor between 51 and 63 is 3. Therefore, GCD(51, 63) = 3. Since the GCD is not 1, 51 and 63 are not coprime.
b) 52, 91
Now, we determine the GCD of 52 and 91:
- The prime factorization of 52 is [tex]\(2^2 \times 13\)[/tex].
- The prime factorization of 91 is [tex]\(7 \times 13\)[/tex].
The common factor between 52 and 91 is 13. Therefore, GCD(52, 91) = 13. Since the GCD is not 1, 52 and 91 are not coprime.
c) 71, 81
Next, we determine the GCD of 71 and 81:
- The prime factorization of 71 is a prime number (71 itself).
- The prime factorization of 81 is [tex]\(3^4\)[/tex].
There are no common factors between 71 and 81. Therefore, GCD(71, 81) = 1. Since the GCD is 1, 71 and 81 are coprime.
d) 81, 99
Finally, we determine the GCD of 81 and 99:
- The prime factorization of 81 is [tex]\(3^4\)[/tex].
- The prime factorization of 99 is [tex]\(3^2 \times 11\)[/tex].
The common factor between 81 and 99 is 3. Therefore, GCD(81, 99) = 3. Since the GCD is not 1, 81 and 99 are not coprime.
Based on our evaluations, the pair (71, 81) is the only pair that is coprime. Therefore, the correct answer is:
C) 71, 81
