How many of the integers between 110 and 120 are prime numbers?

### Concept & Strategy To find prime numbers within a small range, write out the odd integers and systematically eliminate those divisible by 3, 5, and 7. Even integers greater than 2 are never prime. ### Step-by-Step Solution * **List potential primes:** The odd integers between 110 and 120 are: 111, 113, 115, 117, 119. * **Apply Divisibility Rules:** - **111:** Sum of digits is $1 + 1 + 1 = 3$. Divisible by 3. - **115:** Ends in 5. Divisible by 5. - **117:** Sum of digits is $1 + 1 + 7 = 9$. Divisible by 3 (and 9). - **119:** Test divisibility by 7. $119 = 7 \times 17$. It is composite. * **Remaining Number:** - **113:** Its approximate square root is slightly less than 11. It is not divisible by 2, 3, 5, or 7. Therefore, 113 is a prime number. * Only one integer in this range is prime. ### Exam Strategy & Shortcut Instantly filter out the evens and the number ending in 5. You are left checking only 111, 113, 117, and 119. 111 and 117 obviously fall to the rule of 3. You only have to actively remember that 119 is $7 \times 17$. This leaves 113 as the sole prime. ### Common Pitfall Failing to recognize that 119 is a composite number. Because 119 does not fall to the simple divisibility rules of 2, 3, or 5, students frequently count it as a prime, mistakenly choosing option (c) 2. ### Final Answer **Therefore, the correct answer is 1.**