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) $

$\Rightarrow f(2) = 5$

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

$= 24 + 5 + 5 = 34$