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, f(mn) = f(m)f(n)
when m= n= 1, f(1) = f(1)*f(1) ==> f(1) = 1

when m=1,  n= 2, f(2) = f(1)*f(2) ==> f(1) = 1

when m=n= 2, f(4) = f(2)*f(2) ==> f(4) = $$[f(2)]^2$$

Similarly f(8) = f(4)*f(2) =$$[f(2)]^3$$

f(24) = 54

$$[f(2)]^3$$ * $$[f(3)]$$ = $$3^3*2$$

On comparing LHS and RHS, we get 

f(2) = 3 and f(3) = 2

Now we have to find the value of f(18)

f(18) = $$[f(2)]$$ * $$[f(3)]^2$$

= 3*4=12