100 Doors
There are $100$ initially locked doors numbered
\[
1, 2, \ldots, 100
\]
A doorman makes $100$ passes. On pass $k$, he scans every door which is a multiple of $k$. A scan has the following result:
\[
S=
\begin{cases}
\text{locked}, & \text{if unlocked},\\
\text{unlocked}, & \text{if locked}.
\end{cases}
\]
As such, on scan $1$, every door is unlocked. But which doors are unlocked at the end?