To determine which group of numbers are all prime numbers, we need to check each number in the groups for primality.
Step-by-step solution:
1. Group A: [tex]\(2, 5, 15, 19\)[/tex]
- [tex]\(2\)[/tex] is a prime number.
- [tex]\(5\)[/tex] is a prime number.
- [tex]\(15\)[/tex] is not a prime number because it can be divided by [tex]\(3\)[/tex] and [tex]\(5\)[/tex].
- [tex]\(19\)[/tex] is a prime number.
Since [tex]\(15\)[/tex] is not prime, group A does not consist of all prime numbers.
2. Group B: [tex]\(7, 17, 29, 49\)[/tex]
- [tex]\(7\)[/tex] is a prime number.
- [tex]\(17\)[/tex] is a prime number.
- [tex]\(29\)[/tex] is a prime number.
- [tex]\(49\)[/tex] is not a prime number because it is [tex]\(7 \times 7\)[/tex].
Since [tex]\(49\)[/tex] is not prime, group B does not consist of all prime numbers.
3. Group C: [tex]\(3, 11, 23, 31\)[/tex]
- [tex]\(3\)[/tex] is a prime number.
- [tex]\(11\)[/tex] is a prime number.
- [tex]\(23\)[/tex] is a prime number.
- [tex]\(31\)[/tex] is a prime number.
Since all numbers in group C are prime, this group consists of all prime numbers.
4. Group D: [tex]\(2, 3, 5, 9\)[/tex]
- [tex]\(2\)[/tex] is a prime number.
- [tex]\(3\)[/tex] is a prime number.
- [tex]\(5\)[/tex] is a prime number.
- [tex]\(9\)[/tex] is not a prime number because it is [tex]\(3 \times 3\)[/tex].
Since [tex]\(9\)[/tex] is not prime, group D does not consist of all prime numbers.
After checking each group, we find that the group where all numbers are prime is Group C: [tex]\(3, 11, 23, 31\)[/tex].
Therefore, the answer is Group C.
