modular arithmetic - Show that if $n > 0$ is an integer, then $n^2$ is congruent to only $0,1,2$ or $4$ modulo $7$

No solution please, I just need some guidance. I've tried various approaches so far yet no prevail.

• I've looked at small number cases and tried to identify something interesting. Couldn't find much though.


• I've thought about how n can only be odd or even, hence I took the form n = 2k or n = 2k+1 and squared each respectively. Which was indeed interesting because the odd took the form of $2k^2 + 4k + 1$ after being squared which seems interesting because n^2 can be congruent to 2,4 and 1 mod 7. But I can't really advance from there, or if this isn't relevant.

I'm hoping to get a sense of direction from you. Thanks.

Answer

You only have to consider $7$ numbers:

$$0,1,2,3,4=7-3, 5=7-2, 6=7-1$$

Square each of them and you are done.

Another way to phrase it is each number can be written as $7k+r$ where $r \in \{0,1,2,3,4,5,6\}$, just study $(7k+r)^2$ to convince yourselves that it suffices to study $7$ numbers.