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In the Quantum Computation and Quantum Information (10th ed.) textbook by Nielsen and Chuang, section 5.3.1 (titled "Application: order-finding") describes how phase estimation can be used to find the order $r$ of a positive integer.

After performing the inverse Fourier transform on the first register and then performing a measurement, we'll get an estimation of the phase $\varphi\approx\frac{s}{r}$ accurate to $2L+1$ bits. We hope to find $s$ and $r$ by performing the continued fractions procedure, which in certain cases returned $s'$ and $r'$ instead which happens when $s$ and $r$ share common factors.

The book describes three methods to fix this problem, two of which repeats the order-finding algorithm $O(L)$ times and the third requires a constant number of trials.

The third method:

The idea is to repeat the phase estimation - continued fractions procedure twice, obtaining $r'_1$, $s'_1$ the first time, and $r'_2$, $s'_2$ the second time. Provided $s'_1$, $s'_2$ have no common factors, $r$ may be extracted by taking the least common multiple of $r'_1$, $r'_2$

Question:

I don't understand why $r$ may be extracted by taking the least common multiple of $r'_1$, $r'_2$ and what is the rationale of repeating the phase estimation — continued fractions procedure twice?

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Numerators and denominators of convergents of continued fractions are always co-prime. If $s$ and $r$ have a common factor, the $r'$ returned by the continued fractions algorithm would be a factor of $r$.
For example, $r=60$, provided the phase estimation step gives a good approximate to $40/60$ (i.e. $s=40$), you'll get $2/3$ as a convergent. Performing the phase estimation again might give you a good approximate to $21/60$ (i.e. $s=21$), and you'll get $7/20$ as a convergent. $lcm(3,20)=60$ will give you the correct $r$.
Although doing phase estimation-CF twice doesn't guarantee the correctness of resultant $r$ (e.g. if we got $40/60=2/3$ and $28/60=7/15$ in our two attempts, $lcm(3,15)=15$ does not give us the correct $r$), some number theoretic theorem guarantees that we succeed with probability at least $1/4$ (see the last remark on the last page of Prof Jozsa's Part II QI&C Notes).
(P.S. I'll update my answer with the aforementioned theorem of number theory when I find it)

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