Problem

Source: 2023 Serbia TST Problem 5

Tags: nt, Factorials, Weird, Serbia, TST, modular arithmetic, number theory



For positive integers $a$ and $b$, define \[a!_b=\prod_{1\le i\le a\atop i \equiv a \mod b} i\] Let $p$ be a prime and $n>3$ a positive integer. Show that there exist at least 2 different positive integers $t$ such that $1<t<p^n$ and $t!_p\equiv 1\pmod {p^n}$.