Any clues on how to do this modular arithmetic proof? Assume:


*

*$2x^3 - 8x^2 + 8y^3 - 12y^2 -10 \equiv 0 \mod 10$.

*$2y^3 - 8y^2 + 8z^3 - 12z^2 -10 \equiv 0 \mod 10$.


WTP:


*

*$2x^3 - 8x^2 + 8z^3 - 12z^2 -10 \equiv 0 \mod 10$.


I'm not sure where to start. How should I go about this?

Reducing $\pmod {10}$ gives:
$2x^3 +2x^2 - 2y^3 - 2y^2 \equiv 0 \pmod {10}$
So:
$2x^3 +2x^2 \equiv 2y^3 + 2y^2  \pmod {10}$
Similarly from eqn. 2 we get:
$2z^3 +2z^2 \equiv 2y^3 + 2y^2  \pmod {10}$
The equivalence relation is transitive, therefore:
$2z^3 +2z^2 \equiv 2x^3 + 2x^2  \pmod {10}$
and so eqn. 3 is true.
