Let n and n + 2 be two consecutive odd integers. The difference between the squares of these two consecutive odd integers is always divisible by which of the following numbers?

Introduction:
This conceptual number theory question asks about a general pattern involving consecutive odd integers and their squares. Rather than plugging in specific examples, we can use algebra to prove a property that holds for all such pairs of odd numbers.

Given Data / Assumptions:

• We consider two consecutive odd integers. They can be written as k and k + 2 for some integer k.
• We look at their squares: k^2 and (k + 2)^2.
• We must determine a number that always divides the difference between these squares.

Concept / Approach:
The difference of squares formula is crucial: a^2 − b^2 = (a − b)(a + b). For consecutive odd integers, the difference between the two numbers is 2, and their sum is an even multiple of 2, which leads to a factor of 4. Careful simplification reveals that the difference between their squares is always a multiple of 8.

Step-by-Step Solution:
Let the two consecutive odd integers be (2n + 1) and (2n + 3), where n is any integer. Their squares are (2n + 1)^2 and (2n + 3)^2. Compute the difference D = (2n + 3)^2 − (2n + 1)^2. Use the difference of squares formula: D = [(2n + 3) − (2n + 1)] * [(2n + 3) + (2n + 1)]. Simplify the first factor: (2n + 3) − (2n + 1) = 2. Simplify the second factor: (2n + 3) + (2n + 1) = 4n + 4 = 4(n + 1). Thus D = 2 * 4(n + 1) = 8(n + 1). Therefore, the difference between the squares is always 8 times an integer. So D is always divisible by 8.

Verification / Alternative check:
Test with a few examples. Take 3 and 5: 5^2 − 3^2 = 25 − 9 = 16, which is divisible by 8. Take 7 and 9: 9^2 − 7^2 = 81 − 49 = 32, also divisible by 8. These examples support the general algebraic proof.

Why Other Options Are Wrong:
While the difference is also divisible by 2 and 4 (since 8 is a multiple of both), the question asks for a number that always divides the difference, and 8 is the strongest choice here. The option 6 does not always divide the difference; for instance, 16 and 32 are not divisible by 6. Therefore, 8 is the correct and most informative answer.

Common Pitfalls:
Some learners guess from small examples and choose 2 or 4, missing that 8 always works. Others may incorrectly define consecutive odd numbers or forget to use the difference of squares identity. Writing the numbers in the form (2n + 1) and (2n + 3) is a systematic way to avoid mistakes.

Final Answer:
The difference between the squares of two consecutive odd integers is always divisible by 8.