Shourya's function

Algebra Level 5

Function \(f\) from the positive integers to the positive integers satisfies the following conditions:

(1) \(f(xy)=f(x)+f(y)-1\), for any pair of positive integers \(x\) and \(y\).

(2) \(f(x)=1\) holds for only finitely many \(x\).

(3) \(f(30)=4\).

What is the value of \(f(14400)\)?

This problem is posed by Shourya P.

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