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f(xy) = f(x) + f(y) + kf(m_{xy}) where m_{xy}= gcd (x,y)

Source: V- Rioplatense 1996 L3 P6

September 19, 2022

number theorygreatest common divisor

Problem Statement

Find all integers

k

for which, there is a function

f: N \to Z

that satisfies: (i)

f(1995) = 1996

(ii)

f(xy) = f(x) + f(y) + kf(m_{xy})

for all natural numbers

x, y

,where

m_{xy}

denotes the greatest common divisor of the numbers

x, y

.
Clarification:

N = \{1,2,3,...\}

and

Z = \{...-2,-1,0,1,2,...\}