Modulo Division is not generally Well-Defined

Theorem

The operation of division modulo $m$ operation on $\Z_m$, the set of integers modulo $m$, defined by the rule:

$a \div_m b$ equals the integer $q \in \Z_m$ such that $b \times_m q \equiv a \pmod m$

is not in general a well-defined operation.


That is:

If $a \equiv b \pmod m$ and $x \equiv y \pmod m$, then it is not always the case that $a \div_m x \equiv b \div_m y \pmod m$.


Proof


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Sources

  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): congruence (modulo $n$)
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): congruence equation
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