Which of the following is a prime number?

### Concept & Strategy To verify if a number $N$ is prime, you only need to check for divisibility by prime numbers up to the approximate square root of $N$, denoted as $\sqrt{N}$. ### Step-by-Step Solution * **Option (a) 115:** The number ends in 5, making it divisible by 5. Thus, it is composite. $$115 = 5 \times 23$$ * **Option (b) 119:** The approximate square root is slightly less than 11 (since $11^2 = 121$). Test primes: 2, 3, 5, 7. It is not divisible by 2, 3, or 5. Check 7: $119 = 7 \times 17$. It is composite. * **Option (c) 127:** The approximate square root is slightly more than 11. Test primes: 2, 3, 5, 7, 11. It is not even, sum of digits (10) isn't divisible by 3, doesn't end in 5, $127 \div 7$ leaves remainder 1, and $127 \div 11$ leaves remainder 6. Since it isn't divisible by any prime up to its square root, 127 is prime. ### Exam Strategy & Shortcut Quickly eliminate obvious composites using the rules for 2, 3, and 5. For 119 and 127, memorizing multiples of 7 (like $7 \times 10 = 70$, $70 + 49 = 119$) helps you instantly eliminate 119. ### Common Pitfall Assuming 119 is prime because it doesn't immediately look divisible by small numbers like 2, 3, or 5. Forgetting to test the number 7 is the most common mistake here. ### Final Answer **Therefore, the correct answer is 127.**