Question

show that n^2 -1 is divisible by 8 if n is an odd positive integer.

Answer 1

which method binomial theorem or mathematical induction

Answer 2

$we \: know \: \ that \: any \: odd \: positive \: integer \: of \: the \: form \: 4q + 1 \: 4q + 3 \: 4q + 5 \ \\ now \: let \: n \: = \: 4q + 1 \\ then \\ n ^{2} - 1 = (4q + 1)^{2} - 1 \\ = n^{2} - 1 = (16q^{2} + 8q + 1) - 1 \\ = 8(2q^{2} + q) \\ hence \: 8|n^{2} - 1$