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

Concept / Approach

• Let the consecutive odd integers be (2k + 1) and (2k + 3).
• Use the identity a^2 − b^2 = (a − b)(a + b).

Step-by-step derivation(2k + 3)^2 − (2k + 1)^2 = [(2k + 3) − (2k + 1)] × [(2k + 3) + (2k + 1)]= (2) × (4k + 4) = 8(k + 1)

ConclusionThe expression is always a multiple of 8 (but not always 16), hence divisible by 8.

Final Answer8