An Application of Cauchy’s Functional Equation

Cauchy’s functional equation is a description of a function.

$$ f(x + y) = f(x) + f(y) $$

Lets look at Indian National Math Olympiad 2018’sProblem 6 which can be solved as an application of Cauchy’s Functional Equation:

INMO 2018 Problem 6

Let N denote the set of all natural numbers and let \(f : N\rightarrow N\) be a function such that
(a) \(f{(mn)} = f {(m)} f{(n)}\) for all m,n in N ;
(b) m+n divides \(f {(m)} + f {(n)} \) for all m, n in N
Prove that there exists an odd natural number \(k\) such that \(f {(n)} = n^k\) for all n in N.