Congruence Modulo Negative Number

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Theorem

Let $a, b, c \in \R$ be real numbers.

Then:

$a \equiv b \pmod c \iff a \equiv b \pmod {-c}$


Proof

\(\ds a\) \(\equiv\) \(\ds b\) \(\ds \pmod c\)
\(\ds \leadstoandfrom \ \ \) \(\ds \paren {a - b}\) \(=\) \(\ds k c\) Definition of Congruence Modulo $c$: for some $k \in \Z$
\(\ds \leadstoandfrom \ \ \) \(\ds \paren {a - b}\) \(=\) \(\ds -k \paren {-c}\)
\(\ds \leadstoandfrom \ \ \) \(\ds a\) \(\equiv\) \(\ds b\) \(\ds \pmod {-c}\) Definition of Congruence Modulo $-c$

$\blacksquare$


Sources