# How to get common roots of unity of $z^{6}=1$ and $z^{21}=1$?

How to get common roots of unity of $$z^{21}=1$$ and $$z^{6}=1$$ for $$z\in\mathbb C$$?

I know that $$z^{n} =1$$ has roots $$z=e^{\frac{2\pi k }{n}i}$$ where $$k\in \{0,1,2,...,n-1\}$$

• Do you mean $z^6=1?$ – mfl Jan 23 at 0:46
• I have corrected!! – sejy Jan 23 at 1:17
• Find the roots of $z^3$ as $\left(z^3\right)^2=z^6$ and $\left(z^3\right)^7=z^{21}$. – Mohammad Zuhair Khan Jan 23 at 1:20
• Note that $z^{21-3\times 6}=z^3=1$. – lulu Jan 23 at 1:21
• In $z^{21}=1$ for what values k we have $e^{i\pi/3}$?? And how – sejy Jan 23 at 1:42

$$\gcd( 21, 6 ) = ???$$

$$\frac{ 21 }{ 6 } = 3 + \frac{ 3 }{ 6 }$$ $$\frac{ 6 }{ 3 } = 2 + \frac{ 0 }{ 3 }$$ Simple continued fraction tableau:
$$\begin{array}{cccccc} & & 3 & & 2 & \\ \frac{ 0 }{ 1 } & \frac{ 1 }{ 0 } & & \frac{ 3 }{ 1 } & & \frac{ 7 }{ 2 } \end{array}$$  $$7 \cdot 1 - 2 \cdot 3 = 1$$

$$\gcd( 21, 6 ) = 3$$
$$21 \cdot 1 - 6 \cdot 3 = 3$$

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a different example:

$$\gcd( 7654321, 1234567 ) = ???$$

$$\frac{ 7654321 }{ 1234567 } = 6 + \frac{ 246919 }{ 1234567 }$$ $$\frac{ 1234567 }{ 246919 } = 4 + \frac{ 246891 }{ 246919 }$$ $$\frac{ 246919 }{ 246891 } = 1 + \frac{ 28 }{ 246891 }$$ $$\frac{ 246891 }{ 28 } = 8817 + \frac{ 15 }{ 28 }$$ $$\frac{ 28 }{ 15 } = 1 + \frac{ 13 }{ 15 }$$ $$\frac{ 15 }{ 13 } = 1 + \frac{ 2 }{ 13 }$$ $$\frac{ 13 }{ 2 } = 6 + \frac{ 1 }{ 2 }$$ $$\frac{ 2 }{ 1 } = 2 + \frac{ 0 }{ 1 }$$ Simple continued fraction tableau:
$$\begin{array}{cccccccccccccccccc} & & 6 & & 4 & & 1 & & 8817 & & 1 & & 1 & & 6 & & 2 & \\ \frac{ 0 }{ 1 } & \frac{ 1 }{ 0 } & & \frac{ 6 }{ 1 } & & \frac{ 25 }{ 4 } & & \frac{ 31 }{ 5 } & & \frac{ 273352 }{ 44089 } & & \frac{ 273383 }{ 44094 } & & \frac{ 546735 }{ 88183 } & & \frac{ 3553793 }{ 573192 } & & \frac{ 7654321 }{ 1234567 } \end{array}$$  $$7654321 \cdot 573192 - 1234567 \cdot 3553793 = 1$$

$$z$$ will satisfy both of $$z^{21}=1$$ and $$z^6=1$$ iff $$z^3=1$$. One direction is trivial, the other follows from $$z^6=1\implies z^{21}=z^3$$.

Thus $$z\in \{ 1,e^{\frac{2\pi i}3},e^{\frac{4\pi i}3} \}$$.

\begin{align}\{ z : z^6=z^{21}=1, z\in\mathbb C \}&\Longleftrightarrow \{ k : k=o(z),\ k|6,\ k|21 \}\\ &\Longleftrightarrow\{ k : k=o(z),\ k|\gcd (6, 21) \}\\ &\Longleftrightarrow\{ k : k=o(z),\ k|3 \}\\ &\Longleftrightarrow\{ k : k=o(z),\ k=1\ \text{or}\ k=3 \}\\ &\Longleftrightarrow\{ z\in \mathbb C : z^3=1 \}\\ &\Longleftrightarrow\{ z : z=1\ \text{or}\ z=e^{\frac{2\pi i}{3}}\ \text{or}\ z=e^{\frac{4\pi i}{3}} \} \end{align}