Find the minimum value of the function

Putting $m=1$ ,

We get $f(n)=f(1)\times f(n)$

$\Rightarrow f(1)=1$

Since it is a one-to-one function, $f(n)>1$ for all values of $n>1$

Now, $999=3^3\times 37$

$\Rightarrow f(999)=f(3)^3\times f(37)$

Since it is a one-to-one function, $f(3)\geq2$ and $f(37)\geq3$

So, for the least value of $f(999)$ , we claim by setting that $f(2)=37, f(3)=2, f(37)=3$ .

So, the function can be defined as for a prime no. $p\neq2,3,37 f(p)=p$ and for a composite no. it can be prime factorized as above.

This gives $f(999)=2^3\times3=\boxed {24}$