The difference between the squares of two consecutive even integers will always be divisible by which of the following?

(A) 2

(B) 3

(C) 4

(D) 5

Choose the correct answer from the options given below:

Let the two consecutive even integers be $n$ and $n + 2$, where $n$ is even.

The difference between their squares is:

$(n + 2)^2 - n^2$

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Using the difference of squares formula $a^2 - b^2 = (a + b)(a - b)$:

$(n + 2)^2 - n^2 = (n + 2 + n)(n + 2 - n)$

$= (2n + 2)(2)$

$= 4(n + 1)$

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Testing divisibility by 2:

$4(n + 1) = 2 \times 2(n + 1)$

Always divisible by 2.

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Testing divisibility by 3:

$4(n + 1)$ is divisible by 3 only when $(n + 1)$ is divisible by 3.

For $n = 2$: $4(3) = 12$ is divisible by 3.

For $n = 4$: $4(5) = 20$ is not divisible by 3.

Not always divisible by 3.

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Testing divisibility by 4:

$4(n + 1)$ has 4 as a factor.

Always divisible by 4.

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Testing divisibility by 5:

$4(n + 1)$ is divisible by 5 only when $(n + 1)$ is divisible by 5.

For $n = 2$: $4(3) = 12$ is not divisible by 5.

Not always divisible by 5.

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The difference is always divisible by 2 and 4, but not always by 3 or 5.

The answer is (A) and (C) only.