Modulo Multiplication is Well-Defined/Warning

Theorem

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:

$a x \equiv b y \pmod z$

Proof

$\ds a$

$\equiv$

$\ds b$

$\ds \pmod m$

$\ds x$

$\equiv$

$\ds y$

$\ds \pmod m$

$\ds \leadsto \ \ $

$\ds a \bmod m$

$=$

$\ds b \bmod m$

Definition of Congruence

$\ds x \bmod m$

$=$

$\ds y \bmod m$

$\ds \leadsto \ \ $

$\ds \exists k_1 \in \Z: \, $

$\ds a$

$=$

$\ds b + k_1 z$

$\ds \exists k_2 \in \Z: \, $

$\ds x$

$=$

$\ds y + k_2 z$

$\ds \leadsto \ \ $

$\ds a x$

$=$

$\ds \paren {b + k_1 z} \paren {y + k_2 z}$

Definition of Multiplication

$\ds $

$=$

$\ds b y + b k_2 z + y k_1 z + k_1 k_2 z^2$

Integer Multiplication Distributes over Addition

$\ds $

$=$

$\ds b y + \paren {b k_2 + y k_1 + k_1 k_2 z} z$

But it is not necessarily the case that:

$b k_2 + y k_1 + k_1 k_2 z$

is an integer.

In fact, $b k_2 + y k_1 + k_1 k_2 z$ can only be guaranteed to be an integer if each of $b, y, z \in \Z$.

Hence $a b$ is not necessarily congruent to $x y$.

$\blacksquare$

Sources

1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $24$