Which is a prime number 234 OR 788 OR 123 OR 563?

The only even prime is 2, so neither 234 nor 788 is prime. Since 1+2+3=6 is divisible by 3, the divisibility test for 3 says 123 can not be prime. If one of the options must be prime, then we know it is 563. Just to check:

563 is odd and not divisible by 3,5,7,11,13,17,19, or 23. Since it is not divisible by any primes less than its square root, we know that 563 is not divisible by any whole numbers other than itself and one and is hence prime.