Question

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

• A. 31
• B. 32
• C. 33
• D. 34

Answer 1

Answer: (D)

Given, $f(xy)=f(x)+f(y)$

$f(12)=24$

$\Rightarrow f(8)=15$

$f(8)=f(2\cdot 2\cdot 2)=f(2)+f(2)+f(2)$

$\Rightarrow 15=3f(2)$

$\Rightarrow f(2)=5$

$\therefore f(48)=f(12\cdot 2\cdot 2)=f(12)+f(2)+f(2)$

$=24+5+5=34$