Number Sets Analysis

The question asks us to identify which one set of numbers is different from the others based on a specific rule. We are given four sets of numbers:

• Set 1: (3, 5, 19, 13)
• Set 2: (11, 19, 2, 3)
• Set 3: (3, 29, 2, 13)
• Set 4: (2, 31, 11, 4)

Understanding Prime and Composite Numbers

To find the rule, let's recall the definitions of prime and composite numbers:

• Prime Numbers: A natural number greater than 1 that has no positive divisors other than 1 and itself. Examples include 2, 3, 5, 7, 11, 13, 17, 19, etc.
• Composite Numbers: A natural number greater than 1 that is not prime. This means it has at least one divisor other than 1 and itself. Examples include 4, 6, 8, 9, 10, 12, etc.

Analyzing Each Number Set

Let's examine each number in all the given sets to determine if it is a prime or a composite number.

First Number Set: (3, 5, 19, 13)

• 3: Is a prime number (divisors are 1, 3).
• 5: Is a prime number (divisors are 1, 5).
• 19: Is a prime number (divisors are 1, 19).
• 13: Is a prime number (divisors are 1, 13).

Observation: All numbers in the set (3, 5, 19, 13) are prime numbers.

Second Number Set: (11, 19, 2, 3)

• 11: Is a prime number (divisors are 1, 11).
• 19: Is a prime number (divisors are 1, 19).
• 2: Is a prime number (divisors are 1, 2). It is the smallest prime number and the only even prime number.
• 3: Is a prime number (divisors are 1, 3).

Observation: All numbers in the set (11, 19, 2, 3) are prime numbers.

Third Number Set: (3, 29, 2, 13)

• 3: Is a prime number (divisors are 1, 3).
• 29: Is a prime number (divisors are 1, 29).
• 2: Is a prime number (divisors are 1, 2).
• 13: Is a prime number (divisors are 1, 13).

Observation: All numbers in the set (3, 29, 2, 13) are prime numbers.

Fourth Number Set: (2, 31, 11, 4)

• 2: Is a prime number (divisors are 1, 2).
• 31: Is a prime number (divisors are 1, 31).
• 11: Is a prime number (divisors are 1, 11).
• 4: Is a composite number (divisors are 1, 2, 4). Since it has divisors other than 1 and itself (namely 2), it is not prime.

Observation: The number 4 in the set (2, 31, 11, 4) is a composite number, while the other numbers in this set (2, 31, 11) are prime.

Conclusion

Based on our analysis:

• The first three sets (3, 5, 19, 13), (11, 19, 2, 3), and (3, 29, 2, 13) consist exclusively of prime numbers.
• The fourth set (2, 31, 11, 4) contains one composite number, which is 4, along with prime numbers.

Therefore, the set (2, 31, 11, 4) is different from the other sets because it is the only one that includes a composite number.