Data Sufficiency

Hello,

Quick question about DS problem 170 in the OG12:

If n is a positive integer, is n^3 - n divisible by 4?

(1) n = 2k + 1, where k is an integer.
(2) n^2 + n is divisible by 6.

OA: A

I followed through the logic of this, realizing n^3 - n was the series of consecutive integers (n - 1)n(n +1). But I interpreted this to mean that these three could only be divisible by 4 if either n was a multiple of four or if n was any odd number greater than 1.

Because statement (1) says k is an integer, even if it is 1, then n = 3, which makes the statement sufficient. However, I was reading the MGMAT OG Companion, which states "We are told that n must be a positive integer. Thus, the lowest possible value of n - 1 is zero. Even in this case, the product will be divisible by 4, since 0 is divisible by 4."

I don't follow that statement. If statement 1 didn't specify that k was an integer, and therefore n could have been 1, n + 1 could have been 2, and n - 1 could have been 0, statement 1 would have still be sufficient?

It's a bit of a hypothetical, but I want to make sure I understand in case something like that ever shows up. Thanks!