How to approach this? If $2m^3 - 8m^2 + 8n^3 - 12n^2 -10 \equiv 0 \mod 10$, then $8m^3 - 12m^2 + 2n^3 - 8n^2 - 10 \equiv 0 \mod 10$

Question

Assuming

$$2m^3 - 8m^2 + 8n^3 - 12n^2 -10 \equiv 0 \mod 10$$

Prove

$$8m^3 - 12m^2 + 2n^3 - 8n^2 - 10 \equiv 0 \mod 10$$

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I tried the following:

$$8m^3 - 12m^2 + 2n^3 - 8n^2 - 10 \equiv 6m^3 - 4m^2 - 6n^3 + 4n^2 \mod 10$$

I'm not sure where to go from here or even if this is correct.

Answer 1

Let $$A=2m^3 - 8m^2 + 8n^3 - 12n^2 -10 $$ and $$B= 8m^3 - 12m^2 + 2n^3 - 8n^2 - 10$$

Note that $$A+B = 10 m^3-20m^2+10n^3-20n^2-20 \equiv 0 , \mod (10)$$

So if one of $A$ or $B$ is a multiple of $10$ so is the other.