Finding $N$ in the relation $de\equiv 1\bmod N$, given $d$ and $e$

Question

If I have a number $e$ and its multiplicative inverse, $d$, is it possible using the Euclidean Algorithm to get $N$ in the congruence:

$$de \equiv 1 \bmod N$$

Answer 1

In general, $N$ is not unique.

$de-1\equiv 0\pmod N$ implies $N$ is a factor of $de-1$.

For instance with $d=2, e=5$, $N$ could possibly be 9 or 3.

Answer 2

There's not even a unique such $N$. For example, take $3\cdot 3$. That's congruent to 1 mod 2,4, or 8.