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

Question

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

Accepted Answer

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 [f(2)]^2 [ f ( 2 ) ] 2 Similarly f(8) = f(4)*f(2) = [ f ( 2 ) ] 3 [f(2)]^3 [ f ( 2 ) ] 3 f(24) = 54 [ f ( 2 ) ] 3 [f(2)]^3