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Let $f$ be a function such that $f (mn) = f (m) f (n)$ for every positive integers $m$
and $n$. If $f (1), f (2)$ and $f (3)$ are positive integers, $f (1) < f (2),$ and $f (24) = 54$, then $f (18)$ equals _______

Given that, $f(mn) = f(m) f(n) \quad \longrightarrow (1)$

And, $f(24) = 54$

$\Rightarrow f (2 \ast 12) =54$

$\Rightarrow f(2) f(12) =54$

$\Rightarrow f(2) f(2*6) =54$

$\Rightarrow f(2) f(2) f(6) =54$

$\Rightarrow f(2) f(2) f(2\ast 3) = 54$

$\Rightarrow f(2) f(2) f(2) f(3) = 54$

$\Rightarrow \left( {f(2)} \right)^{3} f(3) = 3^{3} \times 2$

On comparing both sides, we get

$\boxed {f(2) = 3, f(3) = 2}$

Therefore, $f(18) = f(2*9) = f(2) f(9)$

$\qquad \qquad = f(2) f(3*3) = f(2) f(3) f(3)$

$\qquad \qquad = 3\ast 2\ast 2 =12$

Correct Answer $: 12$
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