Question

If $p$ and $q$ are distinct prime numbers, then the number of distinct non-real numbers which are $p^{th}$ as well as $q^{th}$ roots of unity are:

• A. Min $(p,q)$
• B. Max $(p,q)$
• C. $1$
• D. $0$

Answer 1

The correct option is D $0$

$x^p=1$
$x_1=e^t$ , where $t=\frac{2ik\pi }{p}$ , where $0\le k<p$
Consider,
$x^q=1$
$x_2=e^u$ , where $u=\frac{2im\pi }{q}$ , where $0\le m<p$
If $x_1=x_2$,
$\frac{2ik\pi }{p}=\frac{2im\pi }{q}$
$kq=mp$.
Now $q$ and $p$ are distinct prime numbers. Hence, the only possible cases are :
a) $k=m=0$. However, this is not possible as it would mean $x_1$ and $x_2$ would be equal to 1 (real).
b) $k=p$ and $m=q$. This is also not possible, as $k<p,m<q$
Hence, there can be no common roots of the $p^{th}$ and $q^{th}$ roots of unity.
Hence, D is the correct option.