The most important first step here is to write out the first several factors of the product, in improper fraction form:
3/2 * 4/3 * 5/4 * 6/5 * 7/6 * 8/7 * 9/8 * 10/9 * ... * (n+2)/(n+1)
How did I know I should use improper fractions, as opposed to mixed fractions (e.g. 1 1/2) or decimals (e.g. 1.5)?
For one thing, I can't think of a single GMAT problem I've ever seen in which mixed fractions were useful, and decimals are very rarely useful. In fact, I've had problems in which the question stem and the answer choices were all in decimal form, and yet I converted the decimals from the question stem into fractions, did the work, and then converted my answer back to decimal form at the end to match the answer choices.
In this particular problem, the word "product" is a big red flag, warning us to avoid using mixed fractions or decimals.
Now, going back to the question: we want to know whether the product is an integer. So, I have to look at the product and wonder: under what circumstances would this product, which includes a bunch of consecutive integers in the numerators and a bunch of consecutive integers in the denominator, be an integer - and under what circumstances would it not be an integer.
Just to rephrase that product in a more convenient form:
(3*4*5*6*7*8*9*10*...*(n+2)) / (2*3*4*5*6*7*8*9*...*(n+1))
Why would that EVER not be an integer??
Well, the denominator starts from 2, which the numerator is missing.
Thankfully, the numerator does have one extra factor at the end, (n+2).
So, if (n+2) contains a '2' in its prime box, it will compensate for that missing '2' at the beginning, giving us a YES answer to the question.
But, if (n+2) doesn't contain a '2' in its prime box (i.e. it's odd), then the answer to the question is NO.
So, rephrase the question: is (n+2) even?
Since consecutive integers alternate between odd an even, a further rephrase could look like this:
Is n even?
Now I'm ready to evaluate the statements.
Start with statement 2, as it tells me precisely what I wanted to know! Eliminate choices ACE.
Statement 2 doesn't provide any information about divisibility by 2.
Since the divisors 2 and 3 share no common factors (they are coprime), information about divisibility by 3 sheds no light about divisibility by 2. Eliminate D and choose B.
_________________