What is the smallest prime number that is greater than 200?

Introduction / Context:
Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. Questions about primes often ask for the smallest or largest prime in a given range. Here, we need the smallest prime number strictly greater than 200.

Given Data / Assumptions:
- We are considering natural numbers greater than 200.
- We must find the first number after 200 that is prime.
- Prime numbers have exactly two distinct positive divisors: 1 and the number itself.

Concept / Approach:
We test successive integers starting from 201 and check whether each is composite or prime. A composite number has at least one divisor other than 1 and itself. We continue until we reach a number that has no such divisors. That number will be the smallest prime greater than 200.

Step-by-Step Solution:
Step 1: Check 201. It is divisible by 3 because the sum of its digits is 2 + 0 + 1 = 3, and 3 divides 201. So 201 is not prime. Step 2: Check 203. It is divisible by 7 because 7 * 29 = 203. So 203 is composite. Step 3: Check 205. It is divisible by 5 because it ends in 5. So 205 is composite. Step 4: Check 207 (not listed as an option but useful for verification). The sum of its digits is 2 + 0 + 7 = 9, and 3 divides 9, so 3 divides 207. Therefore, 207 is composite. Step 5: Check 209. It is 11 * 19, so it is composite. Step 6: Check 211. It does not have any divisors among small primes 2, 3, 5, 7, 11 and 13. None of these divide 211 evenly, so 211 is prime. Step 7: Since 211 is the first prime encountered after 200, it is the smallest prime greater than 200.

Verification / Alternative Check:
To reinforce the result, you may check divisibility of 211 up to the square root of 211, which is a bit more than 14. Prime candidates to test are 2, 3, 5, 7, 11 and 13. We already know 211 is not divisible by these, confirming its primality.

Why Other Options Are Wrong:
Option A (201) is divisible by 3, so it is composite.
Option B (205) is divisible by 5, so it is composite.
Option D (203) is divisible by 7, so it is composite.

Common Pitfalls:
A common mistake is to pick the first odd number greater than 200 without checking for divisors. Another error is to test only divisibility by 2 and 3, ignoring other small primes like 5, 7, 11 and 13. Systematic checking is essential for prime identification.

Final Answer:
The smallest prime number greater than 200 is 211.