Modulo Multiplication is Well-Defined
Theorem
The multiplication modulo $m$ operation on $\Z_m$, the set of integers modulo $m$, defined by the rule:
- $\eqclass x m \times_m \eqclass y m = \eqclass {x y} m$
is a well-defined operation.
That is:
- If $a \equiv b \pmod m$ and $x \equiv y \pmod m$, then $a x \equiv b y \pmod m$.
Proof 1
We need to show that if:
- $\eqclass {x'} m = \eqclass x m$
and:
- $\eqclass {y'} m = \eqclass y m$
then:
- $\eqclass {x' y'} m = \eqclass {x y} m$
We have that:
- $\eqclass {x'} m = \eqclass x m$
and:
- $\eqclass {y'} m = \eqclass y m$
It follows from the definition of residue class modulo $m$ that:
- $x \equiv x' \pmod m$
and:
- $y \equiv y' \pmod m$
By definition, we have:
- $x \equiv x' \pmod m \implies \exists k_1 \in \Z: x = x' + k_1 m$
- $y \equiv y' \pmod m \implies \exists k_2 \in \Z: y = y' + k_2 m$
which gives us:
- $x y = \paren {x' + k_1 m} \paren {y' + k_2 m} = x' y' + \paren {x' k_2 + y' k_1} m + k_1 k_2 m^2$
Thus by definition:
- $x y \equiv \paren {x' y'} \pmod m$
Therefore, by the definition of residue class modulo $m$:
- $\eqclass {x' y'} m = \eqclass {x y} m$
$\blacksquare$
Proof 2
The equivalence class $\eqclass a m$ is defined as:
- $\eqclass a m = \set {x \in \Z: x = a + k m: k \in \Z}$
that is, the set of all integers which differ from $a$ by an integer multiple of $m$.
Thus the notation for multiplication of two residue classes modulo $z$ is not usually $\eqclass a m \times_m \eqclass b m$.
What is more normally seen is:
- $a b \pmod m$
Using this notation:
\(\ds a\) | \(\equiv\) | \(\ds b\) | \(\ds \pmod m\) | |||||||||||
\(\, \ds \land \, \) | \(\ds c\) | \(\equiv\) | \(\ds d\) | \(\ds \pmod m\) | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds a \bmod m\) | \(=\) | \(\ds b \bmod m\) | Definition of Congruence Modulo Integer | ||||||||||
\(\, \ds \land \, \) | \(\ds c \bmod m\) | \(=\) | \(\ds d \bmod m\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds a\) | \(=\) | \(\ds b + k_1 m\) | for some $k_1 \in \Z$ | ||||||||||
\(\, \ds \land \, \) | \(\ds c\) | \(=\) | \(\ds d + k_2 m\) | for some $k_2 \in \Z$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds a c\) | \(=\) | \(\ds \paren {b + k_1 m} \paren {d + k_2 m}\) | Definition of Multiplication | ||||||||||
\(\ds \) | \(=\) | \(\ds b d + b k_2 m + d k_1 m + k_1 k_2 m^2\) | Integer Multiplication Distributes over Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds b d + \paren {b k_2 + d k_1 + k_1 k_2 m} m\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds a c\) | \(\equiv\) | \(\ds b d\) | \(\ds \pmod m\) | Definition of Modulo Multiplication |
$\blacksquare$
Examples
Modulo Multiplication: $19 \times 6 \equiv 11 \times 2 \pmod 4$
\(\ds 19\) | \(\equiv\) | \(\ds 11\) | \(\ds \pmod 4\) | |||||||||||
\(\ds 6\) | \(\equiv\) | \(\ds 2\) | \(\ds \pmod 4\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 19 \times 6 = 114\) | \(\equiv\) | \(\ds 11 \times 2 = 22\) | \(\ds \pmod 4\) |
Modulo Multiplication: $2 \times 3 \equiv -6 \times 15 \pmod 4$
\(\ds 2\) | \(\equiv\) | \(\ds -6\) | \(\ds \pmod 4\) | Congruence Modulo $4$: $2 \equiv -6 \pmod 4$ | ||||||||||
\(\ds 3\) | \(\equiv\) | \(\ds 15\) | \(\ds \pmod 4\) | Congruence Modulo $4$: $3 \equiv 15 \pmod 4$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 \times 3 = 6\) | \(\equiv\) | \(\ds \paren {-6} \times 15 = -90\) | \(\ds \pmod 4\) |
Warning
Let $z \in \R$ be a real number.
Let:
- $a \equiv b \pmod z$
and:
- $x \equiv y \pmod z$
where $a, b, x, y \in \R$.
Then it does not necessarily hold that:
- $a x \equiv b y \pmod z$
Sources
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