Yes, you do need to notice that the question talks about 'consecutive numbers' here.

I'm curious where the explanation is from, because part of it is mathematical nonsense- the part that says "in general, what is true for any odd number is true for all odd numbers, and what is true for any even number is true for all even numbers". That's true if you only consider addition, subtraction and multiplication, but is not true if you consider division (or other operations- finding remainders, say, or averaging even numbers). If you know a and b are both even, and are asked if a/b is even, you are not free to simply choose any pair of even numbers for a and b. If a and b are both even, a/b could be even, odd, or a non-integer:

8/4 = 2
10/4 = 2.5
12/4 = 3