CAT 2019

Slot 2

QA

Question & Solution

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, $f (1) \lt f (2)$, and $f (24)$ = 54, then $f (18)$ equals

Given, $f(mn)$ = $f(m)f(n)$

when $m= n= 1$, $f(1) = f(1) \cdot f(1)$ $\Rightarrow f(1) = 1$

when $m=1$,  $n= 2$, $f(2) = f(1) \cdot f(2) $ $\Rightarrow f(1) = 1$

when $m=n= 2$, $f(4) = f(2) \cdot f(2)$ $\Rightarrow f(4) = [f(2)]^2$

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

$f(24) = 54$

$[f(2)]^3 \cdot [f(3)]$ = $3^3 \times 2$

On comparing LHS and RHS, we get 

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

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

$f(18)$ = $[f(2)] \cdot [f(3)]^2$

= $3 \times 4$ = 12