Clearly, 143 is divisible by 11. So, 143 is not prime. 289 is divisible by 17. So, 289 is not prime. 117 is divisible by 3. So, 117 is not prime. $359<(20)^{2}$ and prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, 19. And, 359 is not exactly divisible by any of them. Hence, 359 is a prime number.
Explanation as extracted from the printed page; notation may be imperfect.