Prime Number Identification

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

We need to check each option to see if it fits the definition of a prime number.

Option 1: 513

Check for divisibility by small prime numbers.

• The sum of the digits is $5 + 1 + 3 = 9$.
• Since 9 is divisible by 3, 513 is divisible by 3 ($513 \div 3 = 171$).
• Therefore, 513 is not a prime number.

Option 2: 323

Check for divisibility by small prime numbers.

• It is not divisible by 2, 3, 5, 7, 11, or 13.
• Check divisibility by 17: $323 \div 17 = 19$.
• Since 323 has factors 17 and 19, it is not a prime number.

Option 3: 715

Check for divisibility by small prime numbers.

• The number ends in 5, so it is divisible by 5 ($715 \div 5 = 143$).
• Therefore, 715 is not a prime number.

Option 4: 571

Check for divisibility by prime numbers up to the square root of 571.

• Calculate the square root: $\sqrt{571} \approx 23.89$.
• We need to check prime divisors up to 23: 2, 3, 5, 7, 11, 13, 17, 19, 23.
• 571 is not divisible by 2 (it's odd).
• Sum of digits ($5+7+1=13$) is not divisible by 3.
• Does not end in 0 or 5, so not divisible by 5.
• $571 \div 7 \approx 81.57$
• $571 \div 11 \approx 51.91$
• $571 \div 13 \approx 43.92$
• $571 \div 17 \approx 33.59$
• $571 \div 19 \approx 30.05$
• $571 \div 23 \approx 24.83$
• Since 571 is not divisible by any prime number less than or equal to its square root, 571 is a prime number.

Conclusion: The only prime number among the options is 571.