Is the sum of integers a and b divisible by 6?

IMO, plugging in values for a and b is the fastest approach in this question. We don’t have to try too many values since this is a YES-NO type of DS questions; we will have to take one set of values to get a YES and another set of values to get a NO.

From statement I alone, a is not divisible by 6.
Let a = 3; if b = 3, a+b = 6 is divisible by 6. If b = 4, a+b = 7 is NOT divisible by 6. Statement I alone is insufficient to find if (a+b) is definitely divisible by 6.
Answer options A and D can be eliminated. Possible answer options are B, C or E.

From statement II alone, (a-b) is divisible by 6.
If a= 9 and b = 3, (a-b) = 6 is divisible by 6. In this case, a+b = 12 is divisible by 6.
On the other hand, if a = 7 and b =1, (a-b) = 6 is divisible by 6. In this case, a+b = 8 is NOT divisible by 6.
Statement II alone is insufficient to find if (a+b) is definitely divisible by 6. Answer option B can be eliminated. Possible answer options are C or E.

Combining statements I and II, we have the following:
From statement I, a is not divisible by 6 and from statement II, (a-b) is not divisible by 6.
We can again use the values of a=9 and b=3 to get a YES while a=7 and b=1 will give us a NO. Even after combining the statements, we don’t have a unique YES or a NO as an answer. Answer option C can be eliminated.

The correct answer option is E.

Learning to recycle values (using the same values instead of looking for new values all the time) will help you save time and effort. Observe how I recycled the values when I combined the statements.

Hope that helps!

(1) a is not divisible by 6.
(2) a - b is divisible by 6.

i went by eliminating cases.

Let's start by 2 : a-b is divisible by 6. we can have a=7 and b=1 > and 8 is not divisible by 6 while a=15 and b=3 would give us a yes. Thus, not sufficient.

while 1definitely on its own is not sufficient, coupeled with 2 does not provide any decisive answer. So E

Theory:
1) if M is divisible by P and N is divisible by P then M+-N is divisible by P
2) if M is not divisible by P and N is divisible by P then M+-N is not divisible by P
3) if M is not divisible by P and N is also not divisible by P then we can not say whether M+-N is divisible by P.
4) when M+N is divided by P, has remainder R then it is not necessary that M-N will have the same remainder. (That is to say, if M +N is divisible by P then it's not necessary that M-N will be divisible by P)

Question:
I. Is clearly insufficient as no information about b (b may or may not be divisible by 6) {using 2) & 3) points from above theory}
II. This doesn't guarantee that a+b is divisible by 6 (from point 4) above theory)

Combining 1 & 2
Not sufficient . (one can see this by combining the inference from 2,3,&4 from above points )