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Euler totient and functional equation

Source: 2024 Turkey EGMO TST P2

February 12, 2024
number theoryfunctionDivisibilityfunctional equation

Problem Statement

Find all functions f:Z+→Z+f:\mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+} such that the conditions
  a)   a-b \mid f(a)-f(b) for all a≠ba\neq b and a,b∈Z+a,b \in \mathbb{Z}^{+}
  b)   f(\varphi(a))=\varphi(f(a)) for all a∈Z+a \in \mathbb{Z}^{+} where φ\varphi is the Euler's totient function.
holds
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