Question

Let $A=\{z_1:z_1^{16}=1,z_1ϵC\},B=\{z_2:z_2^{72}=1,z_2ϵC\}andP=\{z_1{z}_2:z_1ϵA,z_2ϵB\}$ are three sets of complex roots of unity (where C denotes set of complex numbers), then

• A. $n(A\cap B)=8$
• B. $n(A\cap B)=4$
• C. $n\left(P\right)=144$
• D. $n\left(P\right)=72$

Answer 1

Answer: (A), (C)

The correct options are
A

$n(A\cap B)=8$

C

$n\left(P\right)=144$

$\frac{2k_1\pi }{16}=\frac{2k_2\pi }{72}\Rightarrow k_1=\frac{2}{9}{k}_2{k}_2=0,\mathrm{9....63}\to 8values\therefore n(A\cap B)=8e(\frac{2k_1\pi }{16}+\frac{2k_2\pi }{72})i$
As $1^{st},{10}^{th},{19}^{th}$ roots are common
$\therefore k_2=0,1,...8,k_1=0,1,....16$ will give all possible $z_1{z}_2$, that is 144.