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

• A. 12
• B. 14
• C. 17
• D. 22

Answer 1

Answer: (A)

12

Answer 2

Given $f(mn)=f(m)f(n)$ and $f(24)=54,$
$f(24)=2×3×3×3$
$f(2×12)=f(2)f(12)=f(2)f(2×6)=f(2)f(2)f(6)=f(2)f(2)f(2×3)=f(2)f(2)f(2)f(3)=2×3×3×3$
Given that $f(1), f(2)$, and $f(3)$ are all positive integers, by comparison, we get $f(2)=3$ and $f(3)=2$. We can safely take $f(1)=1.$
Now, $f(18)=f(2)(9)=f(2)f(3×3)=f(2)f(3)f(3)=3×2×2=12.$