Let a, b, c be positive integers, suppose 5 divides $a^2+b^2+c^2$ prove that 5 divides at least one of a,b or c

Question

Let a, b, c be positive integers. suppose 5 divides $a^2+b^2+c^2$. prove that 5 divides at least one of a,b or c. I don't really know how to approach this problem. Is there a uniform way to do this that I am not seeing?

Answer 1

We have

$$a^2+b^2+c^2=5k \implies 2c^2=5k \implies c= 5h$$

indeed since $(2,5)=1$ we have that $k$ must contain at least one factor $5$.

Answer 2

Modulo $5$, each of $a^2,$ $b^2,$ and $c^2$ must be equivalent to one of $0,$ $1,$ or $-1 \pmod 5.$ (Of course you should show this if you have not already established it.)

There are only a few ways to add three numbers selected from $\{-1,0,1\}$ to get a multiple of $5.$ Are there any ways to do it with just the numbers $\{-1,1\}$?

The condition $a^2 + b^2 = c^2 $ is not necessary.