consider 1st option:
(x+3)/3 = k(integer)
x = 3*(k - 1)
if k = odd (2m+1), it is divisible by 6
if k = even (2m), it is not divisible by 6.

consider 2nd option:
x + 3 = odd
even + odd = odd
-> x is even = 2p
x can be 4 , 6, 8, 10, 12.
hence, can`t be answered with option B alone.

consider 1 and 2:
x = 3*(k -1)
x = 2*p
form both we can say that x is divisible by 3 and 2.

Hence, the ans C

Mjkourtis wrote:Is integer x divisible by 6?
(1) (x+3)/3 is an integer.
(2) x+3 is odd.

Target question: Is integer x divisible by 6?

Statement 1: (x+3)/3 is an integer.
For (x+3)/3 to be an integer, x+3 must be a multiple of 3.
What are some values of x that satisfy this condition?
Well, x could equal ...-9, -6, -3, 0, 3, 6, 9, 12,...
As you can see, the green values are divisible by 6, and the red values are not divisible by 6
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x+3 is odd
If x+3 is odd, then x must be even.
There are several possible values for x that satisfy this condition. Here are two:
case a: x=4, in which case x is not divisible by 6
case b: x=6, in which case x is divisible by 6
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 + 2
Statement 1 tells us that x can equal ...-9, -6, -3, 0, 3, 6, 9, 12,...
Statement 2 eliminates all of the possible values that are odd (i.e., all of the red values).
So, from the combined statements, we can see that x could equal ... -12, -6, 0, 6, 12,...
As you can see, all possible values of x are divisible by 6
Since we can now answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent