Shortcut for solving $165x \equiv 100 \pmod{285}$
The usual way is to check values of x, but if a shortcut is needed then it is needed to convert the equation into an equality:
$165x = 100 + 285k, \exists k \in \mathbb{Z}$
This brings up a linear Diophantine equation (LDE) with $\gcd$ on r.h.s.
$165x -285k = 100$
=> $33x - 57k = 20$
Comparison with the standard form for LDE : $ax + by = c$
brings up the values of coefficients as : $a = 33, b = -57, c = 20$, with the values $x,y$ found by EEA.
Stuck here, please help.
Making use of Diophantine’s equation, \begin{align} -57 & = (-2)\times33 + 9\\ 33 & = 3\times 9 + 6 \\ 9 & = 1\times 6 + 3 \\ 6 & = 2\times 3 + 0 \end{align}
Since $20$ is not divisible by $3$, we have no solution.