I'm trying to understand a Chinese Remainder Theorem proof, with the base case , and I'm hoping someone can help me in my jam.
Statement
Suppose Then the system of congruences
has a unique solution
Proof
Under conditions
and . So
. So
This has a solution and and the full set of solutions set given by
and
So
This apparently immediately demonstrates that this x solves our system of equations and also shows the solution is unique (modulo mn).
That final deduction confuses me. Can someone explain how demonstrates that this x solves the system and is unique