The difference of the squares of two consecutive odd integers is divisible by which of the following integers?

Question: The difference of the squares of two consecutive odd integers is divisible by which of the following integers?

Solution:
Let the two consecutive odd integers be n and n + 2.

Now, the difference of their squares is:
(n+2)² - n²

Using the identity a² - b² = (a - b)(a + b), we get:
(n + 2 - n)(n + 2 + n)
= 2(2n + 2)
= 4(n + 1)

Since n is an odd integer, n + 1 is even. So we can write:
n + 1 = 2k
Substituting this:
4(n + 1) = 4 × 2k = 8k

So, the expression is always divisible by 8.