How to apply modular division correctly?

As described on Wikipedia:

$$\frac{a}{b} \bmod{n} = \left((a \bmod{n})(b^{-1} \bmod n)\right) \bmod n$$

When I apply this formula to the case $(1023/3) \bmod 7$:

$$\begin{align*} (1023/3) \bmod 7 &= \left((1023 \bmod 7)((1/3) \bmod 7)\right) \bmod 7 \\ &= ( 1 \cdot (1/3)) \mod 7 \\ &= ( 1/3) \mod 7 \\ &= 1/3 \end{align*}$$
However, the real answer should be $(341) \bmod 7 = \mathbf{5}$.

What am I missing? How do you find $(a/b) \bmod n$ correctly?

Answer

$\frac{1}{3}\mod 7 = 3^{-1}\mod 7$

You need to solve below for finding $3^{-1}\mod 7$ :

$$3x\equiv 1\pmod 7$$

Find an integer $x$ that satisfies the above congruence