You're right, you probably just missed one part of choice D. D states that the product is divisible by 3 ONLY when n is odd. For example...4(5)6 is divisible by 3 because of the 6 even though n is not odd, so we know D is not correct. Chose E

KFreudenberg ----

n(n+1)(n+2)

1. If n is EVEN, then n= EVEN, n+1 = EVEN+1 = ODD, n+2=EVEN+2=EVEN, then
n(n+1)(n+2) = EVEN (ODD) (EVEN) = EVEN * (EVEN) = EVEN.

2. If n is ODD, then n=ODD, n+1=ODD+1=EVEN, n+2=ODD+2=ODD then,
n(n+1)(n+2) = ODD (EVEN) (ODD) = EVEN * (ODD) = EVEN.

From the above two we can conclude that n(n+1)(n+2) is always EVEN when "n" is either ODD or EVEN.

So, lets look at each option,
A) even only when n is even
Here, they are telling n(n+1)(n+2) is EVEN only when n is EVEN. That is wrong by the conclusion which we made earlier.
So, this option is WRONG.

B) even only when n is odd
Here, they are telling n(n+1)(n+2) is EVEN only when n is ODD. That is wrong by the conclusion which we made earlier.
So, this option is WRONG.

C) odd whenever n is odd
Here, it says, n(n+1)(n+2) is not at all ODD. So, this option is also WRONG.

D) divisible by 3 only when n is odd
Here, it says, n(n+1)(n+2) is divisible by 3 ONLY when n is ODD. But it is also divisible by 3 when n is EVEN. For example, lets take an example,
n=2 (EVEN) then n(n+1)(n+2) = 2(2+1)(2+2) = 2*3*4. It is divisible by 3 even if n is EVEN.
So, this option is also WRONG.

E) divisible by 4 whenever n is even
Here, it says, n(n+1)(n+2) is divisible by 4 whenever n is EVEN. That doesn't say anything if n is ODD.
The word ONLY makes all difference. So, this option is CORRECT.

HTH, GOOD LUCK,

Thanks,
Rajesh,
Loves GMAT...!!!