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# 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.

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