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Weird NT with even weirder scores

Source: 2023 Serbia TST Problem 5

May 22, 2023

ntFactorialsWeirdSerbiaTSTmodular arithmeticnumber theory

Problem Statement

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}