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  4. If m, n are any two odd positive integers with n < m, then the largest positive integer which divides all the numbers of the type m2 - n2 is

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Q. If $m, n$ are any two odd positive integers with $n < m$, then the largest positive integer which divides all the numbers of the type $m^2 - n^2$ is

Principle of Mathematical Induction

Solution:

Let $m = 2k + 1, n = 2k - 1 \left(k \in N\right)$
$\therefore m^{2} - n^{2} = 4k^{2} + 1 + 4k - 4k^{2} + 4k - 1 = 8k$
Hence, all the numbers of the form $m^{2} - n^{2}$ are always divisible by $8$.