Proving 5 divides a, b, c if 5 divides a² + b² + c²

[AlexChandler]

Homework Statement

Prove the following:
If 5 divides $$a^2 + b^2 + c^2$$ then 5 divides a and 5 divides b and 5 divides c.

Homework Equations

$$5 \mid a \implies a=5k , k \in Z$$

The Attempt at a Solution

My idea is to assume 5 divides $$a^2 + b^2 +c^2$$

also assume that "5 does not divide a" or "5 does not divide b" or "5 does not divide c"

then by the division algorithm, for some integers k,s,t

$$a=5k+1$$ or $$a=5k+2$$ or $$a=5k+3$$ or $$a=5k+4$$

$$b=5s+1$$ or $$b=5s+2$$ or $$b=5s+3$$ or $$b=5s+4$$

$$c=5t+1$$ or $$c=5t+2$$ or $$c=5t+3$$ or $$c=5t+4$$

Now if we square each possibility for a, b, and c and add them together in each possible way, we would find that there is no possible combination that will give that 5 divides a^2 + b^2 + c^2. Then we would see that this is a contradiction so we must have that 5 divides a,b, and c. However, there must be an easier way than to have to manipulate 3^4 (i think) equations. Does anybody see a better way to go about this? or a quicker way to check all of the equations?
Thanks!

 

[micromass]

Are you certain that your theorem is even correct?? Have you tried finding counterexample?

 

[AlexChandler]

You are absolutely right! I misread the problem. It says prove that 5 divides a or 5 divides b or 5 divides c. Thanks! This is much easier! Hah i feel stupid

 

[micromass]

You probably want to try this on your own. But it may be worth to see what kind of remainders a² can have when divided by 5...

 

[AlexChandler]

Yes that is exactly right. I have it figured out now. Thank you