AHSME 1960 P25

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Problem Statement

Let $m$ and $n$ be any two odd numbers, with $n$ less than $m$. The largest integer which divides all possible numbers of the form $m^2-n^2$ is:

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Solution

$$m, n \in \mathbb{Z}, \text{ such that } m, n \equiv 1 \mod 2$$ $$m^2 - n^2 = (m - n)(m + n)$$ $$\text{let } k \; \vert \; (m - n)(m + n)$$ $$\text{Want: } max(k)$$ $$k \; \vert \; ((2a + 1) - (2b + 1))((2a + 1) + (2b + 1))$$ $$k \; \vert \; (2a - 2b)(2a + 2b + 2)$$ $$k \; \vert \; 4(a-b)(a+b+1)$$ $$\text{Either } a-b \text{ or } a+b+1 \text{ is even} \implies max(k) = 4 \times 2 = 8$$