Chinese remainder theorem, how does one justify the existence of the solution, without intuiting it?

Question

Chinese remainder theorem for three equations goes like this:

The system of congruences $$x \equiv a_1 \pmod {n_1}$$ $$x \equiv a_2 \pmod {n_2}$$ $$x \equiv a_3 \pmod {n_3}$$ where $n_1$, $n_2$, $n_3$ are pairwise relatively prime, has a solution (existence of a solution).

The second part of the theorem deals with unicity of the solution.

My book justifies the existence of the solution by taking $x= \sum _{1}^3 {N_i a_i x_i}$ where $N_i = n_1 n_2 n_3 / n_i$ and $x_i$ is the inverse of $N_i$ (modulo $n_i$).

My question is how to prove the existence of the common solution without plugging in this solution provided by my book (which seems pulled out of a hat) ?

Are there some higher level maths that can come up with such a out-of-a-hat solution ?

Just wanted to know.

Thanks

The CRT is I think not a totally obvious fact. But there are some more conceptual ideas about coprimeness that might lead you to this formula. For example, you could observe that it suffices to find solutions to $(1, 0, 0)$, $(0, 1, 0)$ and $(0, 0, 1)$, and then the general solution can be found by combining these. If you sit down and try and find a solution for $(1, 0, 0)$, then $N_1 x_1$ is a fairly natural thing to come up with. This pattern comes up more often - for instance that also motivates this. — Izaak van Dongen, Oct 28 '23 at 15:49
As for higher-level maths: if you know any group theory, you can restate the "existence" part of CRT as follows: "the homomorphism $\Bbb Z\to\Bbb Z/n_1\Bbb Z\times\Bbb Z/n_2\Bbb Z\times\Bbb Z/n_3\Bbb Z$ given by $x\mapsto(x\bmod n_1,x\bmod n_2, x\bmod n_3)$ is surjective". A natural way to show this is to show that a generating set for the codomain lies in the image of $f$, which is exactly what I observed in my other comment. This approach also gives another proof of CRT simply by computing $\ker f$. By the way, Langrange interpolation is also related to a homomorphism very similar to this.. — Izaak van Dongen, Oct 28 '23 at 16:40
A simple explanation of the idea behind the CRT formula (cf. @Izaak's first comment) is explained in this answer. Lagrange interpolation viewed as a special case of CRT is explained in this answer. To answer best we need to know your knowledge level. Do you know any ring theory? — Bill Dubuque, Oct 28 '23 at 17:36
@BillDubuque I know what is a ring, but I don't know any ring theory, but it astonishes me that this and the Lagrange polynoms can intersect arithmetics. You guys actually did answer my question concerning advanced math area that handles existence of the solution. I'll try to read a book on the subject of Ring Theory — niobium, Oct 28 '23 at 18:13
If you study ring theory you'd likely be further astonished by the rich connections on this topic. Generally the CRT solvability criterion works because gcd distributes over lcm, and rings (domains) where this is true can be characterized by many interesting properties, e.g. Gauss's Lemma, various ideal distributive laws, gcd $\times$ lcm law, contains = divides, and many other well-known arithmetical properties. That these all become equivalent in this class of rings is one of the most remarkable confluences in mathematics. @Izaak too. — Bill Dubuque, Oct 29 '23 at 00:52
Can't we make a system of three equations with four unknowns and check whether the determinant of the coefficients of unknowns is not zero? only in this case the system of equation has unique solutions.The first equation can be $x-n_1 y-a_1=0$ and so on. — sirous, Oct 30 '23 at 18:11

Chinese remainder theorem, how does one justify the existence of the solution, without intuiting it?

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