A good reference to the general Chinese Remainder Theorem

Question

I am writing a paper on the topology of the Golomb space and need a good (standard) reference to the following

General Chinese Remainder Theorem. For integer numbers $a_1,\dots,a_n$ and positive integers $b_1,\dots,b_n$ the intersection $\bigcap_{i=1}^n(a_i+b_i\mathbb Z)$ is not empty if and only if $a_i-a_j\in d_{i,j}\mathbb Z$ for any $1\le i<j\le n$, where $d_{ij}$ denotes the largest common divisor of the numbers $b_i,b_j$.

Looking at the internet, I found this paper in which the General Chinese Remainder Theorem is formulated as an exercise and another paper in which this theorem is proved.

But I am suspecting that such General Chinese Remainder Theorem should be proved in some standard (undergraduate) textbook in Number Theory. I need it for a proper reference. Please help!

I understand that this is rather a reference request and not a problem of research level. In case of downvotes I will delete it as soon as will get a proper answer from experts.

Answer 1

It seems that you are after this result which can be found, for example, as Theorem 3.12 in Gareth A. Jones, Josephine M. Jones: Elementary Number Theory, Springer-Verlag, London, 1998. Springer Undergraduate Mathematics Series. (It is in the section 3.5 entitled An extension of the Chinese Remainder Theorem.)

Theorem 3.12. Let $n_1,\ldots,n_k$ be positive integers and let $a_1,\ldots,a_k$ be any integers. Then the simultaneous congruences $$x\equiv a_1 \pmod{n_1}, \ldots, x\equiv a_k \pmod{n_k}$$ have a solution $x$ if and only if $\gcd(n_i,n_j)$ divides $a_i-a_j$ whenever $i\ne j$. When this condition is satisfied, the general solution forms a single congruence class $\bmod n$, where $n$ is the least common multiple of $n_1,\ldots,n_k$.