Odd numbers are integers. I believe one of the properties of odd numbers is that, when divided by 2, it leaves a remainder of 1. But yeah, a lot of the books simply state that odd numbers are not divisible by 2.

As far as the question goes:

Statement 1: n is even:
Simply finding two number with contradictory probabilities (i.e. 2 and 4) which render this statement insufficient is fairly easy to do. Therefore options A and D are off the table.

Statement 2: n/2 is odd:
The first thing that strikes me is that, by claiming n/2 to be odd (and therefore an integer), n must be even. This is the same information provided in Statement 1, and therefore option C is off the table (this is one of the mental things I look for when doing DS questions. It helps me eliminate options).

Now let's consider the statement; we know that the number n is an even number, and that it has four definite factors: n (even), n/2 (odd), 2 (even), 1 (odd). Therefore we have 2 odd and 2 even factors of n. The next part is kind of tricky to explain without drawing it out, but I'll give it a shot anyways.

The number '2' cannot be broken down into further factors. It is possible however, that (n/2) has additional factors. It is provided, however, that n/2 is an odd number; as a result n/2 will only have odd factors (odd numbers can only be products of 2 odd numbers). Any combination of two numbers which have a product of n/2, however, will also create two additional even factors. For example, say X * Y = n/2. Therefore X and Y are two factors, both odd, of the number n. X*2 and Y*2, however, will also become 2 even factors of n, since n/X = Y*2, and n/Y = X*2.

Therefore, the probability is 50%, and the final answer is B.