Why is (-3/77) mod 65 equal to 16? Getting an X for Chinese Remainder Theorem (CRT)
In the "Easy CRT" part of the answer to this problem, the author demonstrates that (-3/77) mod 65 is equal to 16. I don't understand - how is this accurate? I sort of understand the steps, but wouldn't the answer just be 62/77?
Thanks, and I apologize if I've missed something obvious!

The definition of $\frac{1}{x}$ is that $\frac{1}{x}$ is the quantity such that $x \cdot \frac1x = 1$ (which may or may not exist). Therefore
$$ \frac{-3}{77} \equiv 16 \pmod {65} \text{ if and only if } -3 \equiv 77 \cdot 16 \pmod {65} $$
This happens if and only if
$$77 \cdot 16 + 3\equiv 0 \pmod {65}$$
which by definition of congruence mod $65$ says that
$$65 \mid (77 \cdot 16 + 3)$$
which is true because $$ 77 \cdot 16 + 3 = 1235 = 65 \cdot 19. $$
