Sure, here is the translated text:

```
II. (20 points) Find all positive integers $n$ such that
$$
\left[\frac{n}{2}\right]+\left[\frac{n}{3}\right]+\left[\frac{n}{4}\right]+\left[\frac{n}{5}\right]=69 \text {, }
$$

where $[x]$ denotes the greatest integer not exceeding the real number $x$.
```
$$
\begin{array}{l}
\frac{n}{2}+\frac{n}{3}+\frac{n}{4}+\frac{n}{5}-4 \\
<\left[\frac{n}{2}\right]+\left[\frac{n}{3}\right]+\left[\frac{n}{4}\right]+\left[\frac{n}{5}\right] \\
\leqslant \frac{n}{2}+\frac{n}{3}+\frac{n}{4}+\frac{n}{5} . \\
\text { Therefore, } \frac{n}{2}+\frac{n}{3}+\frac{n}{4}+\frac{n}{5}-4 \\
<69 \leqslant \frac{n}{2}+\frac{n}{3}+\frac{n}{4}+\frac{n}{5}
\end{array}
$$

Thus, $53<n<57$. Hence, $n=54,55,56$. Upon inspection, $n=55$ satisfies the condition.