You know that 2 is the only even prime number. However, ther are numbers that are odd, but are not prime... obviously. However, there could also be even numbers in this set. So, you cannot really tell.

The second statement tells you that there are no multiples of 4. Okay, so that leaves out all multiples of four, which are even. But how about multiples of 6. Again, you cannot really tell.

If you take the two statements together, you again cannot tell whether it does or does not have any even numbers. In fact, you cannot tell whether it's all even, all odd, or a mix of the two.

Choose E.

Analyse Question-stem:
This is a YES/NO question.
What do we know about even numbers ... they are all divisible by 2.

Check statements ... which is the easier statement to deal with ? For me Statement 2 is slighly easier, so I will start off with this one, and draw out my BD/ACE answer choice grid:

(2) There are no multiples of 4 in S.
By simply listing numbers we can quickly say that this statement is INSUFF:
no multiples of 4 give us the following numbers: 1, 2, 3, 5, 6, 7, 8 , ... so set S could contain all even numbers, all odd numbers, or a mixture of the two.

So we can cross out BD on our answer choice grid. This leaves us with ACE.

(1) There are no prime numbers in S.
Simply list out the numbers which are not prime: 1, 4, 6, 8 , 9, 10, 12, ... so, again, set S could contain all evens (4, 6, 8 ) or all odds (1, 9) ... so this statement is INSUFF to answer the question.

So we can cross out A from ACE ... this leaves us with answer choices C and E.

(1) and (2) together.
So this gives us the following number list:
1, 6, 9, 10, 15
So set S could contain 1, 9, 15 ... all odd numbers, or set could contain 6 and 10, all even numbers. Again this is INSUFF to answer question.

Cross out C ... this leaves the answer E.

Great systematic approach II. Do you do this type of approach for all DS? Would you mind sharing your step by steap process? Much thanks.