Perfect shuffles needed to restore a deck

A 52-card deck is shuffled by splitting it into the top $26$ cards and the bottom $26$ and interleaving them so that the bottom half's cards end up in odd positions $1,3,\dots,51$ and the top half's cards in even positions $2,4,\dots,52$. What is the least number of such shuffles that returns the deck to its original order?

$52$. Each shuffle sends position $x$ to $2x\bmod 53$, so after $n$ shuffles every card is home iff $2^n\equiv1\pmod{53}$. The order of $2$ modulo $53$ is $52$ ($2$ is a primitive root), so $52$ shuffles are needed.