OK - everyone needs to calm down!
Let's discuss a few key concepts:
1) 0 is an even "uncharged" (i.e. it's neither positive nor negative) integer.
2) In data sufficiency, the statements never contradict each other. However, they can be complementary, i.e. have some solutions in common and some not in common.
3) In DS, the statements are the law; when you evaluate a statement, you think "given that this statement is true, do I have enough information to answer the question?".
Now, to this specific question:
Is x an odd integer?
(1) x + 3 is an even integer.
(2) x/3 is an odd integer.
Step 1: analyze the question.
We see the word "is", we think "yes/no question: a definite "yes" is sufficient, a definite "no" is sufficient, a "sometimes yes/sometimes no" is insufficient".
We need to determine if x is an odd integer. What do we know about x? Absolutely nothing. Could be even, odd or a non-integer (sign is irrelevant for determining oddness/evenness, so we don't care about -/0/+).
Step 2: evaluate the statements.
1) x + 3 is even.
We can translate this just like any other word problem, treating "is" as our equal sign:
x + 3 = even
x = even - 3
Well, an even minus an odd will always be odd; that's a definite "yes" answer: sufficient.
We could have also plugged in numbers - let our even number be 2 and we get:
x = 2 - 3 = -1
which is odd... sufficient.
(Note: on odd/even questions, once you've tried one even or odd number, there's no need to test similar numbers.)
2) x/3 is odd
Again, let's translate to an equation:
x/3 = odd
x = odd * 3
Well, an odd times an odd is always odd - that's a definite "yes" answer: sufficient.
Each statement is sufficient alone, no need to combine, choose (D).
Now, for those confused about the seeming conflict between the statements, there is none.
Conflict only exists when the statements cannot generate a common answer to the question. Here, both statements gave us a "definite yes", so we're fine.
Also, the two statements have at least one numerical solution in common (x=3, for example, satisfies both conditions), so again, no conflict.
