Function problem 1 by Dhaval Furia

$f(24)=f(2^3\times 3)=(f(2))^3\times f(3)=54=27\times 2=3^3\times 2\implies f(2)=3, f(3)=2\implies f(18)=f(3^2\times 2)=(f(3))^2\times f(2)=2^2\times 3=\boxed {12}$ .

Given that $f(mn) = f(m)f(n)$ , then $f(n) = f(1)f(n) \implies f(1) = 1$ , $f(n^2)=f(1)f(n)f(n) = (f(n))^2 \implies f(n^k) =(f(n))^k$ for positive integer $k$ . Then

$\begin{aligned} f(24) & = 54 \\ (f(2))^3f(3) & = 3^3 \cdot 2 \\ \implies f(2) & = 3 \\ f(3) & = 2 \\ \implies f(18) & = f(2)(f(3))^2 = 3 \cdot 2^2 = \boxed {12} \end{aligned}$