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## Homework Statement

Prove that if n is an odd positive integer, then [itex]n^2\,\equiv\,1\,\left(mod\,8\right)[/itex].

## Homework Equations

Theorem:

[tex]a\,\equiv\,b\left(mod\,m\right)[/tex]

if and only if

[tex]a\,mod\,m\,=\,b\,mod\,m[/tex]

## The Attempt at a Solution

Using the theorem above:

[tex]a\,=\,n^2[/tex]

[tex]b\,=\,1,\,m\,=\,8[/tex]

Then I have this:

[tex]n^2\,mod\,8\,=\,1\,mod\,8[/tex]

What do I do next to show this for odd positive n?