Question

State whether the following statement is true or false.

If $\omega$ is a cube root of unity such that ${\omega }^n+{\omega }^{\left(2n\right)}=-1$ then the integral n is in the from of $km+I$ where $(k,I)=(3,1),(3,2)$, $m\in Z$

• A. True
• B. False

Answer 1

The correct option is A True

If $w$ is a cube root of unity then $(1+w^n+w^{2n})=0$ when $n$ is not a multiple of $3$.

So $km+I$ should not be multiple of $3$

Now for $k=3$ and $I=1$;

$km+I=3m+1$, which is not a multiple of $3$.

Also for $k=3$ and $I=2$;

$km+I=3m+2$, which is not a multiple of $3$.

Hence above satement is true.