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Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$ for $p\neq q$ adistinct primes.

Where can I start with this proof? It looks similar to the Chinese Remainder Theorem, but that deals with two different a values.

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    A couple of variables denoted with two different letters may take on two different values, *but not necessarily unless they are explicitly stated to have distinct values.*2012-09-03

3 Answers 3

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Here are few proofs of this ubiquitous result = CCRT = Constant-case CRT = Chinese Remainder Theorem. The first three proofs work for for arbitrary coprime naturals $\rm\,p,q.$

Theorem (CCRT) $\rm\ \ \ \begin{align}&\rm x\equiv a\!\!\pmod{\!p}\\ &\rm x\equiv a\!\!\pmod{\!q}\end{align}\!\iff x\equiv a\pmod{\!pq}\,$ if $\rm\,p,q\,$ are coprime

Proof $\ $ For variety we give a few different proofs.

$\rm(1)\ \ \ x \equiv a\pmod {pq}\:$ is clearly a solution, and the solution is $\color{#C00}{unique}$ mod $\rm\,pq\,$ by CRT.

$\rm(2)\ \ \ p,q\:|\:x\!-\!a\iff lcm(p,q)\:|\:x\!-\!a\:$ by the Universal Property of $\rm lcm$.
$\qquad\! $ Further $\rm\:(p,q)=1^{\phantom{|^|}}\!\!\!\!\iff lcm(p,q) = pq,\,$ by this answer.

$(3)\ \, $ By Euclid's Lemma: $\rm\:(p,q)=1,\ p\:|\:qn =\:x\!-\!a\:\Rightarrow\:p\:|\:n\:\Rightarrow\:pq\:|\:nq = x\!-\!a.$

$\rm(4)\ \, $ [Here $\rm\,p,q\,$ are primes] Since the prime factorization of $\rm\,x\!-\!a\,$ is $\color{#C00}{unique}$, and the prime $\rm\,p\,$ occurs in one factorization, and the distinct prime $\rm\,q\,$ occurs in another, both factorizations are the same up to order, hence contain both $\rm\,p\,$ and $\rm\,q,\:$ hence have $\rm\,pq\,$ as a divisor.

Remark $\ $ This constant-case optimization of CRT arises frequently in practice so is well-worth memorizing, e.g. see some prior posts for many examples.

Quite frequently, $\color{#C00}{uniqueness}\ theorems$ provide powerful tools for proving equalities.

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    @Code Yes, $\rm\:(x,y)\:$ means $\rm\:gcd(x,y)\:$ in number theory (common notation).2012-09-23
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Let $[A,B]=\operatorname{lcm}(A,B)$ and $(A,B)=\gcd(A,B)$

If $p,q$ are different integers, $p\mid(x-a)$ and $q\mid(x-a)\implies [p,q]\mid(x-a)$

We know $[p,q]\cdot (p,q)=p\cdot q$

If $(p,q)=1, [p,q]=p\cdot q$

If $p,q$ are distinct primes, $(p,q)=1$

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Let $y=x-a$. We want to show that if $p$ divides $y$ and $q$ divides $y$ then $pq$ divides $y$.

Since $p$ divides $y$, we have $y=pz$ for some $z$. Thus $q$ divides $pz$. Since $q$ is prime, this implies $q$ divides $p$ or $q$ divides $z$. But $q$ cannot divide $p$, so $q$ divides $z$. Suppose that $z=qw$. Then $y=pqw$.