Understanding Co-prime Numbers

Two integers are considered co-prime (or relatively prime) if their only common positive divisor is 1. In other words, their greatest common divisor (GCD) is 1.

To find out which pair of numbers is co-prime, we need to calculate the greatest common divisor for each given pair. If the GCD of a pair is 1, then that pair is co-prime.

Finding Co-prime Pairs: Checking Options

Let's examine each pair provided in the options:

Option 1: (37, 185)

We need to find the greatest common divisor of 37 and 185.

• 37 is a prime number. Its only positive divisors are 1 and 37.
• Let's check if 185 is divisible by 37. We can perform division: $185 \div 37 = 5$.
• Since 185 is divisible by 37, 37 is a common divisor of both 37 and 185.

The common divisors of 37 and 185 are 1 and 37. The greatest common divisor is 37.

$\text{GCD}(37, 185) = 37$

Since the GCD is 37 (which is not 1), the pair (37, 185) is not co-prime.

Option 2: (42, 84)

We need to find the greatest common divisor of 42 and 84.

• We can notice that 84 is a multiple of 42: $84 = 2 \times 42$.
• This means 42 is a common divisor of both 42 and 84. In fact, since 42 divides 84, 42 is the greatest common divisor.

The common divisors of 42 and 84 include 42. The greatest common divisor is 42.

$\text{GCD}(42, 84) = 42$

Since the GCD is 42 (which is not 1), the pair (42, 84) is not co-prime.

Option 3: (23, 151)

We need to find the greatest common divisor of 23 and 151.

• 23 is a prime number. Its only positive divisors are 1 and 23.
• To find the GCD, we check if 151 is divisible by 23. Let's try dividing 151 by 23: $151 \div 23$. $23 \times 6 = 138$ $23 \times 7 = 161$ 151 is not a multiple of 23. The remainder is $151 - 138 = 13$.
• Since 151 is not divisible by 23, the only common positive divisor of 23 (a prime number) and 151 (a number not divisible by 23) is 1.

The only common divisor of 23 and 151 is 1. The greatest common divisor is 1.

$\text{GCD}(23, 151) = 1$

Since the GCD is 1, the pair (23, 151) is a co-prime pair.

Option 4: (28, 49)

We need to find the greatest common divisor of 28 and 49.

• Let's find the divisors of each number. Divisors of 28 are: 1, 2, 4, 7, 14, 28. Divisors of 49 are: 1, 7, 49.
• The common divisors of 28 and 49 are the numbers that appear in both lists: 1 and 7.
• The greatest common divisor among these common divisors is 7.

The common divisors of 28 and 49 are 1 and 7. The greatest common divisor is 7.

$\text{GCD}(28, 49) = 7$

Since the GCD is 7 (which is not 1), the pair (28, 49) is not co-prime.

Conclusion

Based on our calculations, only the pair (23, 151) has a greatest common divisor of 1. Therefore, this is the co-prime pair among the given options.