Modular multiplicative inverse proof

Question

Does the concept of modular multiplicative inverse require a proof or is it taken as a definition?

Suppose $5/4 \equiv 3$ (mod $7$).

Can that even be written in the standard $a = bq + r$ notation and proven from there?

Answer 1

What the statement means is that $5 \times 4^{-1} \equiv 3 \pmod{7}$.

The modular inverse of $4$ is defined as the number $x$ such that $4x\equiv 1 \pmod{7}$ (if it exists), and it's easy to check that $x=2$ satisfies this.

So $5 \times 4^{-1} \equiv 5 \times 2 \equiv 3 \pmod{7}$.

Alternatively, $5/4$ can be defined as the number that when multiplied by $4$, gives $5$, and $3 \times 4\equiv 12 \equiv 5 \pmod{7}$ as required.

Answer 2

$5/4 \equiv 3\pmod 7$ means

$5 \times 4^{-1} \equiv 3 \pmod 7$ or $5 \times 2 \equiv 3 \pmod 7,$

so $5 \equiv 12 \pmod 7$ (i.e., $5=-1\times7+12) $ or $10 \equiv 3 \pmod 7$ (i.e., $10=1\times7+3),$

but $4^{-1} \pmod 7$ is different from $1/4$ as a real number.