What is the greatest prime number you would need to check to determine whether 5,699 is prime?

Unfortunately, there is no easy way to solve this problem. You would need to list out all of the prime numbers and then start checking one by one if 5699 is divisble by that prime number.

So lets list out the first 15 prime numbers:
2, 3, 5, 7 ,11, 13 ,17, 19, 23, 29, 31, 37, 41, 43, 47

Now we need to see if it is divisible by any of these primes:

5699/2 = 2849.5 Not Divisible
5699/3 = 1899.667 Not Divisible
5699/5 = 1139.8 Not Divisible
5699/7 = 814.143 Not Divisible
....
5699/37 = 154.03 Not Divisible
5699/41 = 139 Divisble

So now we found that 5699 = 41 x 139

Can 139 be factored down anymore since 41 is prime and can not be factored?
Nope, 139 is also a prime.

So the greatest prime that you would need to check to determine if 5699 is a prime is 41 . Before we reached 41, we thought that 5699 was a prime, but after finding 41 was divisible into 5699, we now know 5699 is not a prime.

So your final answer would be: