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Find all functions N -> N with three conditions

Source: Iran Third Round MO 1998, Exam 2, P1

July 1, 2012
functionnumber theory proposednumber theory

Problem Statement

Find all functions f:N→Nf: \mathbb N \to \mathbb N such that for all positive integers m,nm,n,
(i) mf(f(m))=(f(m))2mf(f(m))=\left( f(m) \right)^2, (ii) If gcd⁡(m,n)=d\gcd(m,n)=d, then f(mn)⋅f(d)=d⋅f(m)⋅f(n)f(mn) \cdot f(d)=d \cdot f(m) \cdot f(n), (iii) f(m)=mf(m)=m if and only if m=1m=1.
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