2

Part of 2023 Taiwan Mathematics Olympiad

Problems(1)

An old NT problem

Source: 2023 Taiwan Mathematics Olympiad

Find all positive integers

n

satisfying the following conditions simultaneously:
(a) the number of positive divisors of

n

is not a multiple of

8

; (b) for all integers

x

x^n \equiv x \mod n.

Proposed by usjl