Let $A=\left\{1,2,3,5,8,9\right\}.$ Then the number of possible functions $f : A \to A$ such that $f(m. n) = f\left(m\right). f\left(n\right)$ for every $m ,n \in A$with $m.n\in A$ is equal to______.

f(1) = f(1 x 1) = f(1) x f(1) $\Rightarrow$f(1) = 1
f(9) = f(3x 3) = f(3) x f(3) $\Rightarrow$f(3) = 1, f(9) = 1 or f(3) = 3, f(9) = 9
If f(1) = 1, f(3) = 1, f(9) = 1
Then f(2), f(5), f(8) each mapped in 6 ways  
$\therefore$no.of ways = 6³ = 216
$\therefore$ no.of ways = 216, Total ways = 216 + 216 = 432