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In the book "Quantum Computation and Information" by Nielsen and Chuang, Shor's algorithm is presented with a related probability of success theorem and proof found on page 634, Theorem A4.13. In the last paragraph of this proof, the authors write that:

It is easy to see that $r_j | r$ for each j, and therefore $r_j$ is odd, so $d_j = 0$ for all $i = 1,...,k$.

I understand why the last portion of the statement is true, but I do not understand how we see that $r_j | r$.

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TL;DR This is a consequence of the Chinese Remainder theorem and the fact that in any group if $g^k$ is the identity then the order of $g$ divides $k$.


Setup

We are factoring an odd composite positive integer $N=p_1^{\alpha_1}\dots p_m^{\alpha_m}$ where $p_i$ are distinct primes and $\alpha_i$ are positive integers. We choose $x\in\mathbb{Z}_N^*$ uniformly at random. By the Chinese Remainder theorem this choice corresponds to the choice of $m$ numbers $x_j\in\mathbb{Z}_{p_m^{\alpha_m}}^*$ such that $x=x_j\pmod{p_j^{\alpha_j}}$ for every $j=1,\dots,m$. Let $r$ be the order of $x$ and $r_j$ the order of $x_j$.

Proof that $r_j|r$

By the Chinese Remainder theorem, the map from $\mathbb{Z}_N$ to $\mathbb{Z}_{p_1^{\alpha_1}}\times\dots\times\mathbb{Z}_{p_m^{\alpha_m}}$ defined by

$$ x \mapsto(x_1,\dots,x_m) $$

is a ring isomorphism. Therefore, $x^r=1\pmod N$ implies $x_j^r=1\pmod{p_j^{\alpha_j}}$ for every $j$. Now, by group theory, for any $y\in\mathbb{Z}_{p_j^{\alpha_j}}^*$ if $y^k= 1 \pmod{p_j^{\alpha_j}}$ then the order of $y$ divides $k$. Setting $y:=x_j$, we see that the order of $x_j$ divides $r$.$\square$

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