Step 1: Understanding the Question:
The question asks us to identify the prime number from the given options. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. We need to test each option by checking its divisibility by prime numbers up to the square root of the number.
Step 2: Key Formula or Approach:
Find the approximate square root of the numbers (which is around 23 for numbers near 550).
Test divisibility by all prime numbers up to 23: 2, 3, 5, 7, 11, 13, 17, 19, and 23.
Step 3: Detailed Explanation:
Option A (517): The square root is approx 22.7. Let's check primes. Using the alternating sum rule for 11: $5 - 1 + 7 = 11$, which is divisible by 11. Specifically, $517 = 11 \times 47$. So, it is not a prime.
Option B (559): The square root is approx 23.6. Check primes: 2, 3, 5 fail. 7 fails ($559 = 7 \times 79 + 6$). 11 fails. 13: $559 \div 13 = 43$. So $559 = 13 \times 43$. It is not prime.
Option C (571): The square root is approx 23.8. Check primes up to 23. 2, 3, 5 fail easily. 7 fails ($571 = 7 \times 81 + 4$). 11 fails ($5 - 7 + 1 = -1$). 13 fails ($571 = 13 \times 43 + 12$). 17 fails ($571 = 17 \times 33 + 10$). 19 fails ($571 = 19 \times 30 + 1$). 23 fails ($571 = 23 \times 24 + 19$). Since it is not divisible by any prime up to its square root, 571 is a prime number.
Option D (533): Check primes. Divisible by 13: $533 \div 13 = 41$. So $533 = 13 \times 41$. It is not prime.
Option E (539): Sum of alternating digits: $5 - 3 + 9 = 11$, which is divisible by 11. $539 = 11 \times 49$. It is not prime.
Step 4: Final Answer:
The only prime number among the options is 571.