An Application of Cauchy’s Functional Equation

Cauchy’s functional equation is a description of a function. Lets look at Indian National Math Olympiad 2018’s Problem 6 which can be solved as an application of Cauchy’s Functional Equation:

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

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.