Functions Part 2

Concept:Functional equations define exponential and constant functions here.Explanation:From $f(x+y)=f(x)f(y)$ and $f(1)=7$, for positive integers $n$ we get $f(n)=7^n$.From $g(x+y)=g(xy)$, put $y=1$: $g(x+1)=g(x)$.Thus $g$ has period $1$ on the relevant integer inputs, and since $g(1)=1$, we have $g(x)=1$ for every positive integer $x$.The given sum becomes $\sum\limits_{x=1}^{n}\frac{f(x)}{g(x)}=\sum\limits_{x=1}^{n}7^x=19607$.Using the geometric-series formula, $\sum\limits_{x=1}^{n}7^x=7\frac{7^n-1}{7-1}$.So $7\frac{7^n-1}{6}=19607$.Multiply both sides by $6$: $7(7^n-1)=117642$.Divide by $7$: $7^n-1=16806$.Hence $7^n=16807=7^5$.Therefore $n=5$.Answer:$n=5$, which corresponds to option D, i.e. option D.