I posted this in the number theory forum to no success... so I figured maybe the homework help people would have some input(adsbygoogle = window.adsbygoogle || []).push({});

Let x,y,z be integers with no common divisor satisfying a specific condition, which boils down to

[itex]5|(x+y-z)[/itex] and [itex]2*5^{4}k=(x+y)(z-y)(z-x)((x+y)^2+(z-y)^2+(z-x)^2)[/itex]

or equivalently [itex]5^{4}k=(x+y)(z-y)(z-x)((x+y-z)^2-xy+xz+yz)[/itex]

I want to show that GCD(x,y,z)≠1, starting with the assumption 5 dividing (x+y), (z-y), or (z-x) results in x,y or z being divisible by 5. then it's easy to show that 5 divides another term, implying 5 divides all three.

I run into trouble assuming 5 divides the latter part, [itex]2((x+y)^2+(z-y)^2+(z-x)^2)=((x+y-z)^2-xy+xz+yz)[/itex] and showing the contradiction from that point.

Any hints?

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Divisibility rules and proof by contradiction

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