You should have never resurrected this thread from 2012. Anyway,If you consider "x" as an odd number then if you add any even to x, the number will be odd.

Summing x,x+4,x+12,x+18,x+24

We get 5x+58. Which is not divisible by 3 or 5 and is not even or prime number. So none of the above is the answer.

part of your reasoning is wrong. For instancr, if x = 1, 5x + 58 = 63, which

is divisible by 3. For instance, If x = 3, 5x + 58 = 73, which is prime. Thus,

it *can* happen in those categories.