There is an example in my textbook of how you solve:
x2−4≡0mod21⇔x2−4≡0mod3×7
and then 2 congruences can be formed out of this equation if:
x2−4≡0mod3x2−4≡0mod7
and from these 2 congruences result 2 more congruences, for each:
x−2≡0mod3⇒x1=2x+2≡0mod3⇒x2=1x−2≡0mod7⇒x3=2x+2≡0mod7⇒x4=5
and then 4 systems of linear congruences are formed:
{x≡2mod3x≡2mod7
{x≡2mod3x≡5mod7
{x≡1mod3x≡2mod7
{x≡1mod3x≡5mod7
What is the purpose of these systems? I already have the 4 solutions (x1,x2,x3,x4) of the congruence. Why do I need to form these systems?
Answer
Your ‘4 solutions’ are not solutions modulo 21, but pairs of solutions mod on one hand, \bmod 7 on the other hand.
From these pairs of solutions, you recover solutions modulo 3\times 7 with the Chinese remainder theorem.
Start from the Bézout's relation \;5\cdot 3-2\cdot 7=1. Then the solution corresponding to the pair (\color{red}1\bmod3,\color{red}5\bmod7), for instance, will be
x\equiv\color{red}5\cdot5\cdot 3-\color{red}1\cdot2\cdot 7=61\equiv19\mod 21.
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