Question

Let f be a function defined on the set of all positive integers such that $f\left(xy\right)=f\left(x\right)+f\left(y\right)$ for all positive integers $x,y$. If $f\left(12\right)=24$ and $f\left(8\right)=15$, the value of $f\left(48\right)$ is

• A. $31$
• B. $32$
• C. $33$
• D. $34$

Answer 1

Answer: (D)

$34$

From the given definition of the function, taking $x=12$ and $y=4$,

$f\left(48\right)=f\left(12\right)+f\left(4\right)$ .....($1$)

Now, taking $x=4$ and $y=2$,

$f\left(8\right)=f\left(4\right)+f\left(2\right)$ .....($2$)

Now taking $x=y=2$,

$f\left(4\right)=f\left(2\right)+f\left(2\right)=2f\left(2\right)$ .....($3$)

$\Rightarrow f\left(8\right)=2f\left(2\right)+f\left(2\right)=3f\left(2\right)$ .......(from $\left(3\right)$ and $\left(2\right)$)

$\Rightarrow 15=3f\left(2\right)$

$\Rightarrow f\left(2\right)=5$

$\Rightarrow f\left(4\right)=10$ .....(from $\left(3\right)$)

$\Rightarrow f\left(48\right)=24+10=34$ .....(from $\left(1\right)$)