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(2009 ^ {2009 ^ {2009}}) wanted for NT functional , f(a/b)-|f(b)-f(a)|

Source: OIFMAT I 2010 day 1 p1 - Chilean Math Forum FMAT Olympiad https://artofproblemsolving.com/community/c2484778_oifmat

September 25, 2021
number theoryfunctional

Problem Statement

Let f(n) f (n) be a function that fulfills the following properties:
∙\bullet For each natural n n , f(n) f (n) is an integer greater than or equal to 0 0 . ∙\bullet f(n)=2010f (n) = 2010 , if n n ends in 7 7 . For example, f(137)=2010 f (137) = 2010 . ∙\bullet If a a is a divisor of b b , then: f(ba)=∣f(b)−f(a)∣ f \left(\frac {b} {a} \right) = | f (b) -f (a) | .
Find f(200920092009) \displaystyle f (2009 ^ {2009 ^ {2009}}) and justify your answer.
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