Question

Number of values of $x$ (real or complex) simultaneously satisfying the system of equations
$1+z+z^2+z^3+\cdots +z^{17}=0$ and $1+z+z^2+z^3+\cdots +z^{13}=0$ is

• A. $-1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (A)

$-1$

From given, we have,

$z^{18}-1=0\Rightarrow z^{18}=0$

$z^{14}-1=0\Rightarrow z^{14}=0$

as, $z\ne 1$

$z={\sum }_{m=0}^{17}{l}^{i\frac{2m\pi }{18}}$

$z={\sum }_{n=0}^{13}{l}^{i\frac{2n\pi }{14}}$

$l^{i\frac{2m\pi }{18}}=l^{i\frac{2n\pi }{14}}$

$\frac{2m\pi }{18}=\frac{2n\pi }{14}$

solving the above equation, we get,

$\therefore m=8,n=7$

$\therefore -1$ is the correct answer