MathDB

Let

\mathbb{N}_{\geqslant 1}

be the set of positive integers. Find all functions

f \colon \mathbb{N}_{\geqslant 1} \to \mathbb{N}_{\geqslant 1}

such that, for all positive integers

m

and

n

\mathrm{GCD}\left(f(m),n\right) + \mathrm{LCM}\left(m,f(n)\right) = \mathrm{GCD}\left(m,f(n)\right) + \mathrm{LCM}\left(f(m),n\right).

Note: if

a

and

b

are positive integers,

\mathrm{GCD}(a,b)

is the largest positive integer that divides both

a

and

b

, and

\mathrm{LCM}(a,b)

is the smallest positive integer that is a multiple of both

a

and

b

Problem Statement