What is the probability of (x+3)(x+10) being div by 6?

In case this question is confusing for test takers, I'd point out that it doesn't make sense to pose a DS question in this way, for two reasons.

The first is a logical reason: it's not at all clear what information would be sufficient to answer a DS question set up this way. I can illustrate with a simpler example:

If x is a positive integer less than 10, what is the probability x is divisible by 3?
1. x is even
2. 5 < x < 8

This question makes no sense as a DS question. Using neither of the statements, it appears we can already answer the question: 1/3 of the positive integers less than 10 are divisible by 3, so the answer appears to be 1/3. But then when we use Statement 1 alone, we can also answer the question, but we get a different answer: only one of the four even positive integers less than 10 is divisible by 3, so the answer now appears to be 1/4. But using Statement 2 alone, we can answer the question - now we get 1/2, because x is either 6 or 7, one of which is divisible by 3. But then combining the Statements, x must be 6, and now the answer is 1. Where do we stop? How do we know when we have enough information? There's no way to tell, because posing a DS question in this way is illogical. There is no good answer to this question, so the GMAT could never ask it.

Second, the GMAT will never ask "is y divisible by 6?" if y might not even be an integer. I've seen dozens of prep company questions where that's the supposed "trap" -- where a question asks "is k divisible by 5?" or "is k even?" where the "trap" is that k might not be an integer at all. The people designing those questions seem to be under the impression that if k is a non-integer, the answer to the question "is k divisible by 5?" is "no". But in advanced math (in the subjects known as ring theory and field theory), if you extend the concept of divisibility to real numbers, the answer to that question is "yes". Test takers don't need to know anything about how divisibility is extended beyond the set of positive integers, so I won't explain why that's true, but test takers can confidently ignore any prep questions that try to test divisibility using any numbers other than integers - they're not realistic questions.

Of course, the information required is that x is integer. When x is integer, (x+3) and (x+10) can be treated as consecutive if we are looking at the remainder on division by 3. (x+3) will leave the same remainder as (x+3+3+3) or (x+9).
Now, three case..
1) case 1 - x+9 is divisible by 3 or leaves remainder 0, the term (x+3)(x+10) is divisible by 3.
2) case 2 - x+9 leaves remainder 1 when divided by 3 or , the term (x+3)(x+10) will leave 1*2=2.
3) case 3 - x+9 leaves remainder 2 when divided by 3 or , the term (x+3)(x+10) will leave 2*3=6 or 0.

SO 3 cases out of which 2 cases give us remainder 0. Probability = 2/3.

Also, you cannot take x as integer unless it is given, and GMAT will always give you that x is integer.




Of course, x divisible by 2 means x is even number, and not that it can be 3.6 as per GMAT.
Yes, you are correct that the questions so formed may not be exactly as per the actual questions. But one has to understand that the purpose of these questions is to teach the students not to read anything more than what is given in DS questions, and these questions do send that message very clearly and beautifully.

Although I would defer on the value of such questions, I would agree that actual questions may not be worded the same way.