Question

There is a complex number $z$ with imaginary part $164$ and a positive integer n such that $\frac{z}{z+n}=4i.$ The value of $n$ is

Answer 1

$\frac{z}{z+n}=4i$

Let $z=x+iy\Rightarrow x+164i$

$\frac{x+164i}{x+164i+n}=4i$

$x+164i=4i\times (x+n+164i)$ $i^2=-1$

$x+164i=\left(4\right(x+n)i-656)$

on comparing

x = -656

$4(x+n)=164$

$(n-656)=41$

n = 656 + 41 = 697