  Notice that the square of $9376$ is $87909376$ which has as its rightmost
four digits $9376$. To generalize this remarkable fact, we show that, for each
integer $n\ge 2$, there exists at least one and at most two positive integers
$x$ with exactly $n$-digits in base-$10$ (meaning the leftmost or
$n^{\text{th}}$ digit from the right is non-zero) such that squaring the
integer results in an integer whose rightmost $n$ digits form the integer $x$.
We then generalize the argument to prove that, in an arbitrary number base
$B\ge 2$ with exactly $m$ distinct prime factors, an upper bound is $2^m -2$
and a lower bound is $2^{m-1}-1$ for the number of such $n$-digit positive
integers. For $n=1$, there are exactly $2^m -1$ solutions, including $1$ and
excluding $0$.
