Mathematically, congruence modulo is an equivalence relation. We define:

Equivalently: When working in , any number is congruent to an integer if there exists an integer for which .

Now, let's compare the "discrepancies" in the equivalences you note (which are, in fact, all true):

• Note, indeed, that since

• And again, note that since

It is often customary to express equivalence modulo , by choosing in to be such that . But this choice is simply a representative of all the numbers which belong to the same equivalence class, denoted , :

E.g. If , then one of the following holds:

There are several ways, choose the one you prefer.

\documentclass{article}
\usepackage{amsmath}

\renewcommand{\arraystretch}{1.5} % just to make the lines spread out

\begin{document}

\begin{tabular}{lll}
\verb|$a\equiv b \pmod{n}$| & $a\equiv b \pmod{n}$ \\
\verb|$a\equiv b \mod{n}$|  & $a\equiv b \mod{n}$ \\
\verb|$a\equiv b \pod{n}$|  & $a\equiv b \pod{n}$ \\
\verb|$a\equiv b \bmod{n}$| & $a\equiv b \bmod{n}$ & (wrong)
\end{tabular}

\end{document}

Don't forget the braces: try a\equiv b \pmod 11 or a\equiv b \pmod pq and see why.

The last one is marked “wrong”, because the usage is improper: \bmod should be used for the “modulo” binary operation (the one that is often denoted by % in computing).