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Let $n_{1},\ n_{2},\ n_{3},\ \cdots,\ n_{r}$ be positive integers such that $\gcd(n_{i}, n_{j})=1$ for $1 \le \quad i\neq j \quad \le r$

Then the simultaneous linear congruences $ x\equiv a_i \pmod {n_i} $ for all $1 \le i \le r$ has a solution satisfying all these equations. Moreover the solution is unique modulo $n_1 n_2 n_3 \cdots n_r$.

Proof of Existence. I skip proof of uniqueness. Let $n=n_{1}n_{2}n_{3}\cdots n_{r}$. For each integer $k=1,2,3,\ \cdots,\ r$, let $N_{k} =n_{1}n_{2}n_{3}\cdots n_{k-1}n_{k+1}\cdots n_{r}$ = the product of all the moduli $n_{i}$ with the modulus $n_{k}$ missing.

(1) How can you prefigure to consider this, to start the proof? This product is uncanny?

We are given that $\gcd(n_i, n_j)=1$ for $i\neq j$. By reason of $\gcd(a,b) = \gcd(a,c) =1,$ then $\gcd(a, bc) = 1$. Thence $\gcd(N_k, n_k ) = 1$.

Consider if the linear congruence $N_k x\equiv 1 \pmod{n_k}$ has any solutions.

(2) Where does $N_{k}x\equiv 1 \pmod{n_k}$ hail from? How can you prefigure to consider this?

Since $\gcd(N_k, n_k) = 1$, thence by reason of the Linear Congruence Theorem, the linear congruence $N_{k}x\equiv 1 \pmod {n_k}$ has a unique solution. Dub it $x_k$, thence $$ \color{magenta}{ N_k x_k \equiv 1\pmod{n_k}. \tag{♯} }$$

We construct a solution which satisfies all the given simultaneous linear congruences:

$$ x^* =a_1 N_1 x_1 + a_2 N_2 x_2 + a_3 N_3 x_3 +\cdots + a_r N_r x_r $$

(3) Where does this construction hail from? Can you calculate this?

Let us see if this satisfies the first given linear congruence $ x\equiv a_1 \pmod {n_1} $.

By the above definition of $N_{2},\ N_{3},\ N_{4}, \ldots, N_{r}$ these numbers are multiplies of $n_{1}$.

Therefore $ a_{2}N_{2}x_{2}\equiv 0\pmod {n_1}, a_{3}N_{3}x_{3}\equiv 0 \pmod {n_1} \cdot,\ a_{r}N_{r}x_{r}\equiv 0 \pmod {n_1} $,

Thence if I take $x^*$ to modulus $n_1$, then by cause of $\color{magenta}{(♯)}$ $$\begin{align} x^* & \equiv a_{1}N_{1}x_{1}+a_{2}N_{2}x_{2}+a_{3}N_{3}x_{3}+\cdots+a_{r}N_{r}x_{r} \pmod {n_1} \\ & \equiv a_{1} \color{magenta}{ N_{1}x_{1} }+0+0+\cdots+0 \\ & \equiv a_{1} \color{magenta}{ 1 } \pmod {n_1} \end{align}$$

Hence $x^*$ satisfies the first simultaneous congruence $x\equiv a_1 \pmod {n_1}$. Similarly we can show that the solution constructed satisfies the remaining congruences.

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(1) I agree that the product is not that natural at first sight, and maybe someone will give an intuitive or better interpretation, but here is how I see the proof strategy with an analogy of linear algebra : we try to build the $x^{*}$ that verifies the conditions $x^{*} \equiv a_i \mod{n_i}$. Somehow, these conditions can be seen as "projection" equations : when we take the congruence modulo one of the $n_k$s, we need to satisfy a given equation. The hypothesis on the $gcd$s would be a "freedom" hypothesis in a vector space. Once you see the problem like this, this is natural to try to build something similar to a basis (ie coordinates system), such that we can set easily the wanted values $x^{*} \mod{n_k}$ independently.

The product is the first step toward this goal. We know that the product of all $n_k$ is obviously zero modulo any $n_k$, but by removing one, you build a number $N_k$ that is zero modulo any $n_i$ EXCEPT $n_k$ (because of relative primality, it is even invertible modulo $n_k$). You can find some similar constructions e.g. in Lagrange interpolation, when you build very similarly polynomials $L_k$ that are null at any $b_i$ (where $b_i$ are given real numbers) except $b_k$, and there are several other examples. Here, these $N_k$ act as filters, selecting only one value when taking the modulo $n_k$. Well, that's interesting, because if you take a linear combination of these numbers, you can indeed do something like a decomposition on a basis : let $$y=\sum_{i=1}^{r} a_i N_i$$ , then you can do a "projection" and check that : $$y \equiv a_i N_i \mod{n_i}$$

But you don't know $N_i$'s value modulo $n_i$, and you are not already done. You would like to get rid of this annoying $N_i$'s value :

(2) We do something like a normalization. If we could transform $N_i$'s to a new "basis" $N'_i$ such that $$N'_i \equiv 1 \mod {n_i}$$, this would be perfect. This is exactly what is done : the $gcd$'s hypothesis allow you to find an inverse $x_i$ for $N_i \mod {a_i}$, and doing the multiplication, you get a new "unitary basis" $N'_i = x_i N_i$. Once again, you do quite the same than with Lagrange polynomials, when dividing by $X-b_k$ in $L_k=\frac{\prod_{i \neq k}(X-b_i)}{X-b_k}$ such that $L_k(b_k)=1$. Now you just have to :

(3) Build your solution by decomposing it on the new "coordinate system" you just elaborated $$x^{*} = \sum_{i=1}^{r} a_i N'_i$$

You can indeed calculate this, because the inverse $x_i$ you have to find in (2) is given by the extended Euclid algorithm (that gives the bezout coefficients $u,v$ such that $u n_i + v N_i = 1$, then taking the congruence you see that in fact $v \mod{n_i}$ is a solution). Once you have $x_k$, since you have $n_k$ and $a_k$, nothing prevents you from practically compute $x^{*}$.

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An example might help. Suppose we wish to solve the simultaneous congruences \begin{align} x &\equiv 2 \pmod{8}\\ x &\equiv 5 \pmod{9}\\ x &\equiv 6 \pmod{25}\\ \end{align}

Note that $8, 9,$ and $25$ are pairwise prime and $8 \times 9 \times 25 = 1800$. The CRT states that the mapping $$f:\mathbb Z_{1800} \to \mathbb Z_{8}\times \mathbb Z_{9} \times \mathbb Z_{25}$$ defined by $f(\bar x) = (\bar x, \bar x, \bar x)$ is a group isomorphism. We are specifically seeking an $x$ such that $f(\bar x) = (\bar 2, \bar 5, \bar 6)$.

Because $f$ is an isomorphism, there exists integers $e_1, e_2, $ and $e_3$ such that \begin{align} f(\bar{e_1}) &= (\bar 1, \bar 0, \bar 0)\\ f(\bar{e_2}) &= (\bar 0, \bar 1, \bar 0)\\ f(\bar{e_3}) &= (\bar 0, \bar 0, \bar 1)\\ \end{align}

It follows that $x \equiv 2e_1 + 5e_2 + 6e_3 \pmod{1800}$.

So we need to find the values of $e_1, e_2,$ and $e_3$.

From $(\bar{e_1}, \bar{e_1}, \bar{e_1}) = f(\bar{e_1}) = (\bar 1, \bar 0, \bar 0)$, we conclude that \begin{align} e_1 &\equiv 1 \pmod 8\\ e_1 &\equiv 0 \pmod{9}\\ e_1 &\equiv 0 \pmod{25}\\ \end{align} So $e_1$ must be a multiple of $9$ and of $25$. So, for some $z, \; e_1 = z\cdot9 \cdot 25 = 225z$. Then \begin{align} e_1 &\equiv 1 \pmod 8\\ 225z &\equiv 1 \pmod 8\\ z &\equiv 1 \pmod 8\\ e_1 &\equiv 225 \pmod{1800} \end{align}

Similarly \begin{align} e_2 &\equiv 1 \pmod 9\\ 200z &\equiv 1 \pmod 9\\ 2z &\equiv 1 \pmod 9\\ z &\equiv 5 \pmod 9\\ e_2 &\equiv 1000 \pmod{1800} \end{align}

and \begin{align} e_3 &\equiv 1 \pmod{25}\\ 72z &\equiv 1 \pmod{25}\\ -3z &\equiv 1 \pmod{25}\\ z &\equiv 8 \pmod{25}\\ e_3 &\equiv 576 \pmod{1800} \end{align}

So \begin{align} x &\equiv 2e_1 + 5e_2 + 6e_3 \pmod{1800}\\ x &\equiv 2 \cdot 225 + 5 \cdot 1000 + 6 \cdot 576 \pmod{1800}\\ x &\equiv 450 + 5000 + 3456 \pmod{1800}\\ x &\equiv 8906 \pmod{1800}\\ x &\equiv 1706 \pmod{1800}\\ \end{align}

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