"Suppose that n is an odd number. Prove that n^2 is congruent to 1 modulo 8."

Question

I solved by substituting n with 1, 3, 5, and 7 since they are all odd and remainders of mod 8. I concluded that the equation is true because they all are congruent to 1. However I don' t know if that's the correct way to go about it or if I need to create an equation with the variable n.

Please help! Thank you!

Answer 1

n = 2k+1
n^2 = 4k^2 + 4k + 1 = 4k(k+1) + 1
Now, either k or k+1 is even, so 4k(k+1) is a multiple of 8
so n^2 =1 (mod 8)

Answer 2

Thank you so much!