How many prime numbers are there between 100 to 200?

## Concept & Logic This problem tests your direct knowledge of prime number distribution. While there is no simple formula to calculate the exact count of primes in an arbitrary range, standard competitive exam preparation involves memorizing the frequency of primes up to certain milestones (100, 200, 1000). ## Step-by-Step Solution * **Given:** We need to find the total count of prime numbers specifically existing in the range of 100 to 200. * **Logic:** Let's rely on standard prime distributions. * There are exactly **25** prime numbers between 1 and 100. * There are exactly **46** prime numbers between 1 and 200. * **Calculation:** To find the primes strictly between 100 and 200, we subtract the count of the first 100 numbers from the count of the first 200 numbers. $$\text{Primes (100 to 200)} = \text{Primes (1 to 200)} - \text{Primes (1 to 100)}$$ $$\text{Primes (100 to 200)} = 46 - 25 = 21$$ * For reference, the 21 prime numbers are: 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, and 199. ## Exam Strategy & Shortcut **Do not attempt to count primes manually during the exam.** Checking divisibility for numbers up to 200 takes too long. Memorize these three key facts for number system questions: 1. 1 to 50: **15 primes** 2. 1 to 100: **25 primes** 3. 1 to 200: **46 primes** With this memorized, the answer $46 - 25 = 21$ takes 2 seconds. ## Common Pitfall Attempting to list out the primes manually usually results in accidentally including a pseudo-prime (like 119, which is $7 \times 17$, or 133, which is $7 \times 19$) or missing a valid prime, causing students to incorrectly pick 20 or 22. ## Final Answer Therefore, the correct answer is 21.