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GCD and roots of unity

If we have some roots of unity $\zeta$ and $\rho$, in which $o(\zeta)=a$ and $o(\rho)=b$, can we prove that $o(\zeta\rho)=\operatorname{lcm}[a,b]$? If not, can we prove that this is not true?

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    This can be answered in pretty much the same way that your [previous question](http://math.stackexchange.com/questions/72513/gcd-and-roots-of-unity) was answered; did you try to adapt the argument? Is this homework, like the previous one was?2011-10-17
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    @johnnymath: You were not "really" dealing with the gcd in the previous problem. When $\gcd(a,b)=1$, then $\mathrm{lcm}(a,b)=ab$, so the previous problem is **this** problem with the added condition that $a$ and $b$ are relatively prime. Try using Gerry's argument there (but with $0\lt s \lt \mathrm{lcm}(a,b)$) to show $(\zeta\rho)^s\neq 1$, and then show $(\zeta\rho)^{\mathrm{lcm}(a,b)} = 1$.2011-10-17
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    Thanks, now I see the connection2011-10-17

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