Which of the following is a prime number?

### Concept & Formula To determine if a larger number $N$ is prime, find the integer $k$ such that $k > \sqrt{N}$. Then, test $N$ for divisibility by all prime numbers less than or equal to $k$. ### Step-by-Step Solution * **Option (a) 143:** Applying the alternating digit sum rule for 11: $(1 + 3) - 4 = 0$. So, 143 is divisible by 11. $$143 = 11 \times 13$$ * **Option (b) 289:** You should recognize this as a perfect square. $$289 = 17^2$$ * **Option (c) 117:** Sum of the digits is $1 + 1 + 7 = 9$. It is divisible by 9 (and 3). $$117 = 9 \times 13$$ * **Option (d) 359:** Approximate square root: $\sqrt{361} = 19$. Check primes up to 19: 2, 3, 5, 7, 11, 13, 17. 359 is not divisible by any of these primes. Thus, 359 is a prime number. ### Exam Strategy & Shortcut Rely on memorized perfect squares (up to 30) and basic divisibility tricks. 289 is instantly eliminated as $17^2$. 117 is eliminated by the rule of 9. 143 is instantly recognized as $11 \times 13$. Process of elimination leaves 359 as the only possible answer without requiring heavy calculation. ### Common Pitfall Getting bogged down trying to prove 359 is prime manually. It is much faster to prove that 143, 289, and 117 are composite through quick elimination tests. ### Final Answer **Therefore, the correct answer is 359.**