Congruence Modulo Negative Number
Jump to navigation
Jump to search
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
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-1}$ Basic Properties of Congruences