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Let $f: \mathbf{N} \rightarrow \mathbf{N}$ be a function such that $f(m+n)=f(m)+f(n)$ for every $m, n \in \mathbf{N}$. If $f(6)=18$ , then $f(2) \cdot f(3)$ is equal to :
Option: 1 $54$
Option: 2 $6$
Option: 3 $36$
Option: 4 $18$

Answers (1)

best_answer

$f(m+n)=f(m)+f(n),f(6)=18\\$

$Put\; m=n=3;\\$

$f(6)=f(3)+f(3)=18\\$

$\Rightarrow f(3)=9\\$ ....(1)

$Put\; m=4,n=2\\$

$f(6)=f(4)+f(2)=18\\$

Also put $m=n=2\\$

$\Rightarrow f(4)=f(2)+f(2)=2f(2)\\$

$\Rightarrow f(6)=f(2)+f(2)+f(2)=18\\$

$\Rightarrow 3f(2)=18\Rightarrow f(2)=6\\$ ..........(2)

$f(2)\cdot f(3)=9\times6=54$

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Kuldeep Maurya

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