Question

$\text{Find the number of non-zero integral solutions of the equation }|1-i|^x=2^2.$

Answer 1

$|1-i|^x=2^2$

$⇒(\sqrt1^2+(-1)^2)=2^x$

$⇒(\sqrt2)^x=2^2$

$⇒2^{\frac{x}{2}}=2^x$

$⇒\frac{x}{2}=x$

$⇒x=2x$

$⇒2x-x=0$

$⇒x=0$

$\text{Thus, 0 is the only integral solution of the given equation. Therefore, the number of non-zero integral solutions of the given equation is 0.}$