PRMO 2012 question 16

Q16. Let N be the set of natural numbers. Suppose f : N -> N is a function satisfying the following conditions:
(a) f(mn) = f(m)f(n)
(b) f(m) < f(n) if m < n
(c) f(2) = 2

What is the value of $$\sum_{k=1}^{20} f(k)?$$

Answer: 210


Solution:
Since f(1) < f(2), f(1) = 1 (the only natural number < 2).
f(4) = f(2).f(2) = 4
f(2) < f(3) < f(4) so f(3) = 3 is the only choice.
f(6) = f(3).f(2) = 6 so f(5) = 5.

It's clear that f(n) = n is the function definition here.
So we just to have to compute the sum of first 20 natural numbers.

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