I am writing a paper on the topology of [the Golomb space](https://dml.cz/bitstream/handle/10338.dmlcz/700933/Toposym_01-1961-1_41.pdf) 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](http://www.ias.ac.in/article/fulltext/reso/020/03/0206-0216) in which the General Chinese Remainder Theorem is formulated as an exercise and [another paper](http://www.cs.xu.edu/math/math302/08f/06_CRT.pdf) 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. 



A good reference to the general Chinese Remainder Theorem
It seems that you are after this result which can be found, for example, as [Theorem 3.12](https://books.google.com/books?id=LN6PBAAAQBAJ&pg=PA60) 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.)



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> **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 $n\_1,\ldots,n\_k$.
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