TITLE: How to solve $100x +19 =0 \pmod{23}$
QUESTION [5 upvotes]: How to solve $100x +19 =0 \pmod{23}$, which is $100x=-19 \pmod{23}$ ?
In general I want to know how to solve $ax=b \pmod{c}$.

REPLY [7 votes]: Hint $\displaystyle\rm\,\ mod\ 23\!:\,\ x\,\equiv\, \frac{\color{brown}{-19}}{4\cdot \color{#0A0}{25}}
\,\equiv\,\frac{\color{brown}4}{4\cdot\color{#0A0} 2}
\,\equiv\, \frac{\color{blue}1}2
\,\equiv\, \frac{\color{blue}{24}}2
\,\equiv\, 12,\ \:$ 
by $\:\ \begin{array}{r}\color{brown}{{-}19\equiv 4}, & \color{blue}{1\equiv 24}\\ \color{#0A0}{25\equiv 2}\,\end{array} $