IMO 1992 LL COL11

Let arphi(n, m), m ̸= 1, be the number of positive integers less

imolonglistmathematicsolympiad

IMO 1992 LL COL11

Origin: COL

Problem

Let \varphi(n, m), m ̸= 1, be the number of positive integers less than or equal to n that are coprime with m. Clearly, \varphi(m, m) = \varphi(m), where \varphi(m) is Euler’s phi function. Find all integers m that satisfy the following inequality: \varphi(n, m) n \geq\varphi(m) m for every positive integer n.