Question

If a complex number $Z$ with $Im\left(Z\right)=4$ satisfy $\frac{Z}{Z+n}=4i$, where $n$ is a positive integer, then the value of $n$ is

Answer 1

Let $Z=x+iy$
Given $Im\left(Z\right)=4$, then
$Z=x+4i$
Now,
$\frac{Z}{Z+n}=4i\Rightarrow \frac{x+4i}{x+4i+n}=4i\Rightarrow x+4i=4i(x+4i+n)\Rightarrow x+4i=4ix-16+4in\Rightarrow x+4i=-16+4i(x+n)$
Comparing the real and imaginary part, we get
$x=-16\text{and}4=4(x+n)\therefore n=17$