What is the smallest integer $n$ for which both $\pi n$ and $e n$ are within $\frac{1}{10,000}$ of an integer?
I have managed to find some where both are within $\frac{1}{1000}$ of an integer, such as $1,892,803$:
$\pi\cdot 1,892,803=5,946,415.99949$ and $e\cdot 1,892,803=5,145,171.99975$.
Can you find any integers $n$ that makes these numbers within $\frac{1}{10,000}$th of an integer? How about $\frac{1}{100,000}$th?
Here is a systematic way to search for such an integer:
First, look for an integer $r$ such that $r\pi$ is within $1/100,000,000$ of an integer. This search will terminate with $r \le 100,000,000$.
Next, look for an integer $s$ such that $rse$ is within $1/10,000$ of an integer. This search will terminate with $s \le 10,000$.
Now take $n=rs$.
Unfortunately, you asked for the smallest such $n$; and this method is unlikely to find that. But it's a start.