Show that a^2~0 or a^2~1 in Mod 4

[PsychonautQQ]

Homework Statement

If a I san integer, show that a^2~0 or a^2~1 in mod 4 (~ represent equivalence)

Homework Equations

The Attempt at a Solution

my ATTEMPT:
I started with the division algorithm..
a = 2q + 1 for all odd numbers
a = 2q + 0 for all even numbers
then I squared the formula for even numbers...
a^2 = 4q^2 + 0... but q is arbitrary so
a^2 ~ 0 mod 4 for even numbers

then I did the same for odd numbers
a^2 = 4q^2 + 4q + 1
and then I said since q was arbitrary again I could say
a^2 = 4q + 1
a^2~1 mod 4 for odd numbers


is this the correct approach?

 

[1MileCrash]

Yep, clean work.