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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1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 2.6$. Algebra of congruences: $\text{(iii)}$
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2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): congruence (modulo $n$)