Identifying Coprime Numbers

Two numbers are said to be coprime or relatively prime if their greatest common divisor (GCD) is 1. This means that 1 is the only positive integer that divides both numbers.

To find if a pair of numbers is coprime, we need to calculate their GCD. We can do this by listing the factors of each number or by using the prime factorization method.

Analyzing the Given Pairs to Find Coprime Numbers

Let's examine each pair of numbers provided in the options:

Option 1: (7, 49)

• Factors of 7: 1, 7
• Factors of 49: 1, 7, 49

The common factors are 1 and 7. The greatest common divisor (GCD) of 7 and 49 is 7.

Since $\text{GCD}(7, 49) = 7$, which is not 1, the pair (7, 49) is not coprime.

Option 2: (17, 25)

• Factors of 17: 1, 17 (17 is a prime number)
• Factors of 25: 1, 5, 25

The only common factor is 1. The greatest common divisor (GCD) of 17 and 25 is 1.

Since $\text{GCD}(17, 25) = 1$, the pair (17, 25) is a pair of coprime numbers.

Option 3: (8, 52)

• Factors of 8: 1, 2, 4, 8
• Factors of 52: 1, 2, 4, 13, 26, 52

The common factors are 1, 2, and 4. The greatest common divisor (GCD) of 8 and 52 is 4.

Since $\text{GCD}(8, 52) = 4$, which is not 1, the pair (8, 52) is not coprime.

Option 4: (13, 91)

• Factors of 13: 1, 13 (13 is a prime number)
• Factors of 91: 1, 7, 13, 91 (Since $13 \times 7 = 91$)

The common factors are 1 and 13. The greatest common divisor (GCD) of 13 and 91 is 13.

Since $\text{GCD}(13, 91) = 13$, which is not 1, the pair (13, 91) is not coprime.

Conclusion

Based on the analysis of the GCD for each pair:

• (7, 49): $\text{GCD} = 7$
• (17, 25): $\text{GCD} = 1$
• (8, 52): $\text{GCD} = 4$
• (13, 91): $\text{GCD} = 13$

The only pair with a GCD of 1 is (17, 25). Therefore, (17, 25) is a pair of coprime numbers.