Mathematics - Functions Question with Solution | TestHub

Given,

$f\left(mn\right)=f\left(m\right)·f\left(n\right)$

Put $m=1,$ then $f\left(1·n\right)=f\left(1\right)·f\left(n\right)\Rightarrow f\left(1\right)=1 \left\{f\left(n\right)\ne 0\right\}$

Put $m=n=2$

$f\left(4\right)=f\left(2\right)·f\left(2\right)$

If $f\left(2\right)=1,$ then $f\left(4\right)=1$ and if $f\left(2\right)=2,$ then $f\left(4\right)=4$

Put $m=2,n=3$

$f\left(6\right)=f\left(2\right)·f\left(3\right)$

If $f\left(2\right)=1,$ then $f\left(3\right)$ can be any value between $1$ to $7$

If $f\left(2\right)=2,$ then $f\left(3\right)$ can be $1$ or $2$ or $3$

Similarly $f\left(5\right),f\left(7\right)$ can take any value from $\left\{1,2,...,7\right\}$

So, total number of function $=\left(1\times 1\times 7\times 1\times 7\times 1\times 7\right)+\left(1\times 1\times 3\times 1\times 7\times 1\times 7\right)$

$=490$