Which of these is a prime number 3123, 219, 573, 467 :

Options

A. 219
B. 467
C. 573
D. 3123

Answer

To determine which of the given numbers (3123, 219, 573, 467) is a prime number, we need to check each number to see if it has any divisors other than 1 and itself. A prime number is defined as a number greater than 1 that has no positive divisors other than 1 and itself.

### Step-by-Step Solution:

1. **Check 3123**:
- To check if 3123 is prime, we need to see if it can be divided by any number other than 1 and 3123 itself.
- We can check for divisibility by small prime numbers (2, 3, 5, 7, etc.).
- **Divisibility by 3**: The sum of the digits (3 + 1 + 2 + 3 = 9) is divisible by 3. Therefore, 3123 is divisible by 3.
- Since it has divisors other than 1 and itself, **3123 is not a prime number**.

2. **Check 219**:
- Check if 219 is prime by testing divisibility.
- **Divisibility by 3**: The sum of the digits (2 + 1 + 9 = 12) is divisible by 3. Therefore, 219 is divisible by 3.
- Since it has divisors other than 1 and itself, **219 is not a prime number**.

3. **Check 573**:
- Check if 573 is prime by testing divisibility.
- **Divisibility by 3**: The sum of the digits (5 + 7 + 3 = 15) is divisible by 3. Therefore, 573 is divisible by 3.
- Since it has divisors other than 1 and itself, **573 is not a prime number**.

4. **Check 467**:
- Check if 467 is prime by testing divisibility.
- We will check divisibility by prime numbers up to the square root of 467 (approximately 21.6).
- Check divisibility by 2: 467 is odd, so it is not divisible by 2.
- Check divisibility by 3: The sum of the digits (4 + 6 + 7 = 17) is not divisible by 3.
- Check divisibility by 5: 467 does not end in 0 or 5, so it is not divisible by 5.
- Check divisibility by 7: 467 ÷ 7 ≈ 66.71 (not an integer).
- Check divisibility by 11: 467 ÷ 11 ≈ 42.45 (not an integer).
- Check divisibility by 13: 467 ÷ 13 ≈ 36.69 (not an integer).
- Check divisibility by 17: 467 ÷ 17 ≈ 27.47 (not an integer).
- Check divisibility by 19: 467 ÷ 19 ≈ 24.68 (not an integer).
- Since 467 is not divisible by any prime numbers up to its square root, **467 is a prime number**.

### Conclusion:
The only prime number among the given options is **467**.