Logic or meaning of $5 \equiv \frac{4}{8}\pmod {12}$

Question

In modular division, what is the meaning that should be ascribed to the notation exemplified below (also given on p. 5 of this)?

$$\begin{align} \implies & 5\cdot8 \equiv 4\pmod {12} \tag{i} \\[2ex] \implies & 5 \equiv \frac{4}{8}\pmod {12} \tag{ii} \\[2ex] \implies & 8 \equiv \frac{4}{5}\pmod {12} \tag{iii} \end{align}$$

I think in terms of values reached by different residue classes, but I am unable to get any clue.
As a very simple example,
values taken by $4 \pmod{12}$ residue class are: $4, 16, 28, 40$
values taken by $5 \pmod{12}$ residue class are: $5, 17, 29, 41$
values taken by $8 \pmod{12}$ residue class are: $8, 20, 32, 44$

This lends no meaning to eqns. $\text{(ii), (iii)}$ above.

Answer 1

• $5 \cdot 5 =25 \equiv 1 \pmod{12}$ so the multiplicative inverse of $5 \bmod 12\,$ is $5^{-1}=5$.

  Therefore $5 \cdot 8 \equiv 4 \implies 5^{-1}\cdot5\cdot8 \equiv 5^{-1} \cdot 4 \implies 8 \equiv 5^{-1}\cdot 4 \pmod{12}\,$. The latter may sometimes be written as $8 \equiv \frac{4}{5} \pmod{12}\,$ but that's arguably an abuse of notation, unless such notation was very explicitly and narrowly defined before being used.

• $8$ has no multiplicative inverse $\bmod 12$, so $5 \equiv \frac{4}{8} \pmod{12}$ makes no sense whatsoever.

Answer 2

Note that $3*8=24\equiv 0\pmod{12}$. Adding this equation to your equation $(i)$ yields $5*8\equiv 8*8=4\pmod{12}$. So we will get $\frac48\equiv5$ as well as $\frac48\equiv8$ (both mod 12) which is nonsensical (that is not well-defined).

So one should avoid assigning any meaning to division by something which is a zero divisor.

When operating modulo a given $m$, if we stick to $b$ with gcd$(b,m)=1$ all zero divisors will be avoided and fractions with such $b$ in the denominators will be meaningful (will have unique value).