The number of prime numbers between 301 and 320 are

### Concept & Strategy To count the prime numbers in a specific range, list the odd numbers and systematically eliminate multiples of 3, 5, and then test the remaining numbers against larger primes (7, 11, 13, 17). ### Step-by-Step Solution * **List potential primes:** The odd numbers between 301 and 320 are: 301, 303, 305, 307, 309, 311, 313, 315, 317, 319. * **Eliminate obvious composites:** - Multiples of 5 (ends in 5): **305, 315**. - Multiples of 3 (sum of digits is divisible by 3): $303$ ($3+0+3=6$), $309$ ($3+0+9=12$). * **Test remaining numbers (301, 307, 311, 313, 317, 319):** - **301:** Divisible by 7 ($301 = 7 \times 43$). - **307:** Not divisible by 7, 11, 13, or 17. **(Prime)** - **311:** Not divisible by 7, 11, 13, or 17. **(Prime)** - **313:** Not divisible by 7, 11, 13, or 17. **(Prime)** - **317:** Not divisible by 7, 11, 13, or 17. **(Prime)** - **319:** Divisible by 11 ($319 = 11 \times 29$). * **Count the primes:** There are exactly four prime numbers: 307, 311, 313, and 317. ### Exam Strategy & Shortcut Quick elimination is key. You can instantly cross out 303, 305, 309, and 315. For the rest, knowing the divisibility rule for 11 (alternating sum) quickly eliminates 319 ($3 - 1 + 9 = 11$). Recognizing that $301 = 300 + 1$ isn't helpful, but seeing it as $280 + 21$ immediately shows it is a multiple of 7. This leaves only the true primes to verify. ### Common Pitfall Mistakenly counting 301 or 319 as prime numbers because they do not fall to the basic divisibility rules of 2, 3, or 5. Always remember to test division by 7 and 11 for numbers up to 400. ### Final Answer **Therefore, the correct answer is 4.**