How do you write congruence modulo(mod n) in LaTeX?

Congruence modulo syntax will consist of two individual commands, \equiv and \mod commands.

\documentclass{article}
\usepackage{mathtool}
\begin{document}
  % Use mathtools for \mod
  $$ b \equiv c \mod{m} $$
  % \pmod is default command
  $$ a \equiv b \pmod{n} $$
  $$ a \equiv b \pmod{\frac{m}{(k,m)}} $$
  $$ a \equiv b \left(\mod{\frac{m}{(k,m)}}\right) $$
\end{document}

Output :

But, you notice the output above, where a lot of space has been created by using the \mod and \pmod commands.

To solve this problem you need to use \bmod command, or manually solve using \mathrm command.

\documentclass{article}
\begin{document}
 $$ b \equiv c\;(\bmod{m}) $$
 $$ 10+5 \equiv 3\;(\bmod{12}) $$ 
 $$ \frac{p}{q} \equiv f\prod^{n-1}_{i=0}p_i\;(\mathrm{mod}\;m) $$
\end{document}

Output :

Some congruence modulo proparties in LaTeX

Best practice is shown by discussing some properties below.

\documentclass{article}
\usepackage{mathabx}
\begin{document}
\begin{enumerate}
  \item  Equivalence: $ a \equiv \modx{0}\Rightarrow a=b $ 
  \item  Determination: either $ a\equiv b\; \modx{m} $ or $ a \notequiv b\; \modx{m} $ 
  \item  Reflexivity: $ a\equiv a \;\modx{m} $.
  \item  Symmetry: $ a\equiv b\; \modx{m}\Rightarrow b\equiv a \;\modx{m} $.
 \end{enumerate}
\end{document}

Output :

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