https://file.aiquickdraw.com/mathbot/file/1726292762522jpyeifbp.jpg

Let's analyze the sets of numbers in this problem and determine if they are pairwise relatively prime.

### Definition: 
A set of integers is **pairwise relatively prime** if every pair of distinct numbers in the set has a greatest common divisor (GCD) of 1. In other words, no two numbers in the set share a common factor other than 1.

#### Set (a): {11, 15, 19}
- 11 and 15: GCD(11, 15) = 1 (11 is prime and doesn't divide 15)
- 11 and 19: GCD(11, 19) = 1 (both are prime numbers, and 11 and 19 don't divide each other)
- 15 and 19: GCD(15, 19) = 1 (19 is prime and doesn't divide 15)

Since the GCD of every pair of distinct integers is 1, the integers in this set are **pairwise relatively prime**.

#### Set (b): {12, 17, 31, 37}
- 12 and 17: GCD(12, 17) = 1 (17 is prime and doesn't divide 12)
- 12 and 31: GCD(12, 31) = 1 (31 is prime and doesn't divide 12)
- 12 and 37: GCD(12, 37) = 1 (37 is prime and doesn't divide 12)
- 17 and 31: GCD(17, 31) = 1 (both are prime numbers)
- 17 and 37: GCD(17, 37) = 1 (both are prime numbers)
- 31 and 37: GCD(31, 37) = 1 (both are prime numbers)

Since the GCD of every pair of distinct integers is 1, the integers in this set are also **pairwise relatively prime**.

### Conclusion:
- Set (a): **Yes**, they are pairwise relatively prime.
- Set (b): **Yes**, they are pairwise relatively prime.

Would you like any further details on the solution?

Here are some related questions to consider:
1. What is the definition of relatively prime numbers?
2. How do you compute the GCD of two numbers?
3. Can you give an example of a set of numbers that are not pairwise relatively prime?
4. Why are prime numbers often pairwise relatively prime with non-multiples?
5. Can a set of numbers be pairwise relatively prime if one number is divisible by another?

**Tip**: When checking if numbers are pairwise relatively prime, using the Euclidean algorithm to compute the GCD of pairs can be an efficient method.

Determine whether the integers in each of these sets are pairwise relatively prime. (a) 11, 15, 19 (b) 12, 17, 31, 37

Learn how to determine if sets of integers such as {11, 15, 19} and {12, 17, 31, 37} are pairwise relatively prime. This problem covers number theory concepts like the greatest common divisor (GCD) and relatively prime numbers. Suitable for Grades 10-12.

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