Question 44
Let $p$ be a prime number and let $s$ be an integer with $0 \lt s \lt p$. Prove that there exist integers $m$ and $n$ with $0 \lt m \lt n \lt p$ and
$$
\{\frac{sm}{p}\} \lt \{\frac{sn}{p}\} \lt \frac{s}{p}
$$
if and only if $s$ is not a divisor of $p-1$.
(For $x$ a real number, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of $x$.)
$$
\{\frac{sm}{p}\} \lt \{\frac{sn}{p}\} \lt \frac{s}{p}
$$
if and only if $s$ is not a divisor of $p-1$.
(For $x$ a real number, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of $x$.)