expression with inverses in modular equation

Question

How to solve for $x \bmod m$ in :

$$((10210+18371\cdot 8642\cdot x)\cdot (x+2055)^{-1}) \equiv 4804 \mod {20983} $$

And I've found $$(18371\cdot 8642) \equiv 4804 \mod{20983}$$

Answer 1

So you have

$$((10210+4804\cdot x)\cdot (x+2055)^{-1}) \equiv 4804 \mod {20983} \\ \implies 10210+4804\cdot x \equiv 4804(x+2055) \mod {20983} \\ \implies 10210 \equiv 4804\cdot 2055 \mod 20983$$

Which, as it happens, is true. So any value of $x$ will satisfy the equivalence, provided that $(x+2055)^{-1} \bmod 20983$ exists, which since $20983$ is prime, will be true unless $x\equiv -2055 \equiv 18928 \bmod 20983$.