Let f be a function defined on set of all positive integers such that f(x y)=f(x)+f(y) for all posit

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1.Relation and Function

hard

Let $f$ be a function defined on the set of all positive integers such that $f(x y)=f(x)+f(y)$ 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

$34$

(KVPY-2016)

Solution

(d)

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

$f(12)=24 \Rightarrow f(8)=15$

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

$\Rightarrow \quad 15=3 f(2) \Rightarrow f(2)=5$

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

$=24+5+5=34$

Standard 12

Mathematics

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