To determine which group of numbers contains all prime numbers, we need to check each number within each group to see if they are prime.
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
Let's analyze each group:
Group A: [tex]\(2, 3, 5, 9\)[/tex]
- [tex]\(2\)[/tex] is prime (divisors: 1, 2)
- [tex]\(3\)[/tex] is prime (divisors: 1, 3)
- [tex]\(5\)[/tex] is prime (divisors: 1, 5)
- [tex]\(9\)[/tex] is not prime (divisors: 1, 3, 9)
Since [tex]\(9\)[/tex] is not a prime number, Group A does not contain all prime numbers.
Group B: [tex]\(3, 11, 23, 31\)[/tex]
- [tex]\(3\)[/tex] is prime (divisors: 1, 3)
- [tex]\(11\)[/tex] is prime (divisors: 1, 11)
- [tex]\(23\)[/tex] is prime (divisors: 1, 23)
- [tex]\(31\)[/tex] is prime (divisors: 1, 31)
All numbers in Group B are prime. Therefore, Group B contains all prime numbers.
Group C: [tex]\(7, 17, 29, 49\)[/tex]
- [tex]\(7\)[/tex] is prime (divisors: 1, 7)
- [tex]\(17\)[/tex] is prime (divisors: 1, 17)
- [tex]\(29\)[/tex] is prime (divisors: 1, 29)
- [tex]\(49\)[/tex] is not prime (divisors: 1, 7, 49)
Since [tex]\(49\)[/tex] is not a prime number, Group C does not contain all prime numbers.
Group D: [tex]\(2, 5, 15, 19\)[/tex]
- [tex]\(2\)[/tex] is prime (divisors: 1, 2)
- [tex]\(5\)[/tex] is prime (divisors: 1, 5)
- [tex]\(15\)[/tex] is not prime (divisors: 1, 3, 5, 15)
- [tex]\(19\)[/tex] is prime (divisors: 1, 19)
Since [tex]\(15\)[/tex] is not a prime number, Group D does not contain all prime numbers.
Conclusion: Among the given groups, Group B: [tex]\(3, 11, 23, 31\)[/tex] is the group where all the numbers are prime. Therefore, the answer is B.
