Mathematical Induction Proof: n^2 - 1 is divisible by 8 for all odd positive integers

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Mathematical Induction Proof: n^2 - 1 is divisible by 8 for all odd positive integers

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Комментарии
Автор

you don't even need induction to prove the statement just do it directly
suppose n is an integer
(2n+1)^2 -1 = 4n(n+1) = 4 x 2a = 8a, where a is an integer and 2a=n(n+1) since an integer times the next integer will be even). Done

melrobles
Автор

n^2-1 is divider by 8 when n is odd is equivalent to saying that 8 divides (2m-1)^2-1 where mεZ, let’s expand this

=4m(m-1)
Now since If m is odd m-1 is even and inversely if m is even m-1 is odd. Either way there is always one of them that is odd and therefore we can factor 2 out while keeping m(m-1) an integer
4m(m-1)=8m(m-1)/2~0mod(8)

Just another way to solve it(hopefully I’m right and there isn’t a huge flaw in my proof)

eat_your_cereal