Reference for theorem similar to Chinese remainder theorem

Question

I have the theorem below, similar to the Chinese remainder theorem, written in some old notes of mine during my undergraduate degree and have a proof for it, but I want to use it in some work now and would rather avoid writing out a full proof. I am looking for a book or some other material I can refer to with this result in it. I have looked at several number theory books but have been unable to find it stated.

A system of $ r$ linear congruences $$\begin{align*} x &\equiv b_{1}\pmod{n_1}\\ x &\equiv b_{2}\pmod{n_2}\\ &\vdots\\ x &\equiv b_{r}\pmod{n_r}\\ \end{align*}$$ has a simultaneous solution if and only if $ hcf( n_{i} , n_{j} ) $ divides $ b _{j} - b _{i} $ for each pair $ i , j \in \{ 1, \dots , r \} $. Furthermore a solution is unique modulo $ lcm ( n_{1} , n_{2} , \dots , n _{r} ) $ if it exists.

Answer 1

I tried to find a source to a long time ago and was unable to find a book that has it. One option is to reference the proof at this stackexchange thread. As the user ttt states in response to the proof,

This is written up nicely and I can't seem to find it anywhere else

So others have tried to find it and failed. I'm writing a book at the moment that includes a proof. If a difference source is not posted here, I can respond to this thread with a comment when it's available, if you like. It won't be published for another year or so though.

EDIT: Almost three years later, the books have been published (online, available for free). For a proof of this theorem, see Volume 3: Number Theory, Corollary 6.8 (p.79-81). Here is the link to the books.

Answer 2

As Arturo Magidin has suggested, this fact is found in "An introduction to the theory of numbers" by Niven, Zuckerman and Montgomery, as problem 23 in section 2.3 (The Chinese Remainder Theorem) of the 5th edition (1991).