Square of Modulo less One equals One
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Theorem
Let $m \in \Z$ be an integer.
Let $\Z_m$ be the set of integers modulo $m$:
- $\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \ldots, \eqclass {m - 1} m}$
Then:
- $\eqclass {m - 1} m \times_m \eqclass {m - 1} m = \eqclass 1 m$
where $\times_m$ denotes multiplication modulo $m$.
Proof
\(\ds \eqclass {m - 1} m \times_m \eqclass {m - 1} m\) | \(=\) | \(\ds \eqclass {\paren {m - 1}^2} m\) | Definition of Modulo Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds \eqclass {m^2 - 2 m + 1} m\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \eqclass 1 m\) |
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Exercise $2.5$