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In the book Introduction to Analytic Number Theory written by Tom M. Apostol, there is a step in the proof of Theorem 5.26 (Chinese remainder theorem) that I am unable to understand. In the proof, we assume $ m_1, \dots, m_r $ to be positive integers relatively prime in pairs and $ M = m_1 \cdots m_r $. There is a step that proves the uniqueness of solution that I cannot understand. Here it is:

But it is easy to show that the system has only one solution mod M. In fact, if $x$ and $y$ are two solutions of the system we have $x \equiv y \pmod{m_k}$ for each $k$ and, since the $m_k$ are relatively prime in pairs, we also have $x \equiv y \pmod{M}$.

The author does not explain why must $x \equiv y \pmod{m_k}$ for each $k$ imply that $x \equiv y \pmod{M}$?

I know we can show this ourselves. When $gcd(a,b) = 1$, $a|n$ and $b|n$ implies $ab|n$. There are numerous proofs of this available online. This is equivalent to stating that $ n \equiv 0 \pmod{a}$ and $n \equiv 0 \pmod{b}$ implies $n \equiv 0 \pmod{ab}$ when $gcd(a,b)=1$. But is this so obvious that it does not require a proof of its own in the book?

So what I am asking here is how do we justify this step from the preceding material in the book:

we have $x \equiv y \pmod{m_k}$ for each $k$ and, since the $m_k$ are relatively prime in pairs, we also have $x \equiv y \pmod M$

Does the book explain why this should be true somewhere earlier? I could not find anything. If you have read this book and you found the necessary groundwork for this in the preceding material, please share it here. If the book does not explain it somewhere earlier, does this step not require a proof of its own?

Lone Learner
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  • By the linked dupes: universal property of lcm (or CCRT) $\Rightarrow m_1,\ldots, m_r\mid n!\iff! {\rm lcm}(m_1,\ldots,m_r)\mid n,,$ and lcm = product for pairwise coprimes. OP is case $, n = x-y\ \ $ – Bill Dubuque May 12 '21 at 07:56
  • Why is this question marked as a duplicate? I am well aware that the uniqueness can be proved if we search the Internet or this stackexchange. That is not the point of this question. The point is whether the proof is present in the book and if not, if sufficient groundwork has been laid out in the book to make the proof so obvious that the author chose to omit it. The point of this question is to show that the proof of uniqueness of solution is obvious from the preceding material in the book. – Lone Learner May 12 '21 at 10:14
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    The post has a few questions (should have only one). I chose the first (the only mathematical question), viz. "Why does ... how do we justify...?". You can find all of the common proofs by chasing the links I gave. Only the author can tell you which he intended if that was not explicitly specified (but you don't even name the author). Questions about whether or not a proof is in some book are not on-topic imo. It is quite common for authors to omit justifications of claims they consider "obvious to the reader". It is usually of little import which of many equivalent proofs one chooses. – Bill Dubuque May 12 '21 at 17:07

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Sorry, I don't have that book, so I don't know if it has that proof, but here's a very short one.

$a|n$ and $b|n$, so $\frac{n}{a}$ and $\frac{n}{b}$ are integers.

Since $gcd(a,b) = 1$, then there exist $x, y$ such that $$ax + by = 1$$ Hence $$\left(\frac{n}{ab}\right)ax + \left(\frac{n}{ab}\right)by = \frac{n}{ab}$$ So $$\left(\frac{n}{b}\right)x + \left(\frac{n}{a}\right)y = \frac{n}{ab}$$ The LHS is an integer, so $ab|n$.

PM 2Ring
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