Complicated divisibility problem

[ripcity4545]

Homework Statement

If 5 divides m^2 + n^2 + p^2 , prove that 5 divides wither m, or n, or p.

Homework Equations

m,n,p are all integers

The Attempt at a Solution

I am having some major problems with this chapter on modular arithmetic. any help is much appreciated!

modular arithmetic is not needed to solve the problem, but may be helpful.

 

[Tedjn]

What possible values can m^2 have mod 5?

 

[ripcity4545]

What possible values can m^2 have mod 5?

Well if we are considering divisibility by 5, i can let
m=5k+r, where k is some integer and r may be 0, 1, 2, 3, or 4.

So m^2 = 25k^2 + 10kr +r^2
so
m^2 ≡ n mod 5 equals:

25k^2 + 10kr +r^2 ≡ n mod 5
and since 25k^2 + 10kr ≡ 0 mod 5, we are left with
r^2 ≡ n mod 5

so n may be 1, 4, or 0. Am I going in the right direction? Thanks for the reply.

 

[Tedjn]

That's correct. So what can m^2 + n^2 + p^2 be if none are congruent to 0 mod 5?

 

[ripcity4545]

thanks for your help.