JEE Main 2022 · 25 Jul · Shift 1, Mathematics question 21

Structure. $A^2=A$, so $A^n=A$; $B=A-I$ gives $B^2=I-A=-B$, so $(\omega B)^n=\omega^n(-1)^{n-1}B$.

Condition. $\omega^n(-1)^{n-1}=1$, i.e. $(-\omega)^n=-1$ with $-\omega=e^{-i\pi/3}$: $n\equiv3\pmod6$, so $n=3,9,\dots,99$: 17 values.

Answer: 17