Formal Modulus Proof: How close am I?

Show that if n is an odd positive integer then n^2 = 1(mod 8).

I see that any odd square has 1 as a remainder when calculated. Example: 49 = 7 * 7 = 1(mod 4), and any odd number squared equals an odd number. Let 2k represent all positive even integers. So n^2 = 1(mod 2k) for all odd positive integers.

Is this an acceptable proof?

Did you not answer your own question?

Doesn't the theorem explicitly state that the expression must be an integer?

Theorem:

Let *m* be a positive integer. The integers *a *and* b* are congruent modulo *m* if and only if there is an integer *k* such that *a = b + km. *

I definately should have referred to the definition first before trying to solve the problem; I would have had a much easier time. Lesson learned.