My question is about modular inverses and fractions modulo $p$
So I was revising my contest notes, I was trying to think about each topic as if I was explaining it to a child, which is when I realised I did not actually understand what fractions modulo $p$ meant:
I understand what something modulo $p$ means, it essentially tells us the remainder when that thing is divided by $p$.
But what does a fraction modulo $p$ even mean?
Like I can say $9 \equiv 2 \mod(7)$ , cause $9 = 7 \cdot 1 + 2$.
But what does it mean to say $\frac{2}{3} \mod (7)$.
I know how to calculate this, it is just $2 \cdot 3^{-1} \equiv 2 \cdot 5 \equiv 10 \equiv 3 \mod(7)$.
But I want to know what it means. Like intuitively, what does this represent? Like what does it mean to say $\frac23 = (0.6666....) \equiv 3 \mod(7)$
If you were to explain it to a layman, how would you, like you'd say $9 \equiv 2 \mod(7)$ because that is the remainder on division by 7. But how would you explain to someone that $\frac23$ is $3$ modulo $7$?
What I think it may mean is that if $\frac sr \equiv a \mod (p)$, then $p$ would need to divide the numerator of $\frac sr - a$.
Thank you!
