  We show that given the order of a single element selected uniformly at random
from $\mathbb Z_N^*$, we can with very high probability, and for any integer
$N$, efficiently find the complete factorization of $N$ in polynomial time.
This implies that a single run of the quantum part of Shor's factoring
algorithm is usually sufficient. All prime factors of $N$ can then be recovered
with negligible computational cost in a classical post-processing step. The
classical algorithm required for this step is essentially due to Miller.
