- #1

AlexChandler

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## Homework Statement

Prove the following:

If 5 divides [tex] a^2 + b^2 + c^2[/tex] then 5 divides a and 5 divides b and 5 divides c.

## Homework Equations

[tex] 5 \mid a \implies a=5k , k \in Z [/tex]

## The Attempt at a Solution

My idea is to assume 5 divides [tex] a^2 + b^2 +c^2[/tex]

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

[tex] a=5k+1 [/tex] or [tex] a=5k+2 [/tex] or [tex] a=5k+3 [/tex] or [tex] a=5k+4 [/tex]

[tex] b=5s+1 [/tex] or [tex] b=5s+2 [/tex] or [tex] b=5s+3 [/tex] or [tex] b=5s+4 [/tex]

[tex] c=5t+1 [/tex] or [tex] c=5t+2 [/tex] or [tex] c=5t+3 [/tex] or [tex] c=5t+4 [/tex]

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!

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