The number of common roots among the 12^{text | vedclass

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(D) The $n^{	ext{th}}$ roots of unity are given by $e^{i \frac{2k\pi}{n}}$ for $k = 0, 1, 2, \dots, n-1$. For $12^{	ext{th}}$ roots of unity,the val

Vedclass · Accepted Answer

$6$

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(D) The $n^{	ext{th}}$ roots of unity are given by $e^{i \frac{2k\pi}{n}}$ for $k = 0, 1, 2, \dots, n-1$. For $12^{	ext{th}}$ roots of unity,the values are $e^{i \frac{2k_1\pi}{12}} = e^{i \frac{k_1\pi}{6}}$ where $k_1 \in \{0, 1, \dots, 11\}$. For $30^{	ext{th}}$ roots of unity,the values are $e^{i \frac{2k_2\pi}{30}} = e^{i \frac{k_2\pi}{15}}$ where $k_2 \in \{0, 1, \dots, 29\}$. $A$ root is common if $e^{i \frac{k_1\pi}{6}} = e^{i \frac{k_2\pi}{15}}$,which implies $\frac{k_1}{6} = \frac{k_2}{15} \pmod{2}$. This simplifies to $\frac{k_1}{2} = \frac{k_2}{5} \pmod{2}$,or $5k_1 = 2k_2 \pmod{60}$. The number of common roots between $n^{	ext{th}}$ and $m^{	ext{th}}$ roots of unity is given by $\gcd(n, m)$. Here,$\gcd(12, 30) = 6$. Thus,there are $6$ common roots.

The number of common roots among the $12^{\text{th}}$ and $30^{\text{th}}$ roots of unity is

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