Let
and
be two relatively prime elements of an Euclidean domain
.
(You may think
or
.)
Let
be such that
.
For every
there exists
such that
![$\displaystyle (\forall x \in R) \ \ \ \ \ \left\{ \begin{array}{l} x \equiv a \...
...uiv b \mod{\, n} \\ \end{array} \right. \ \ \iff \ \ x \equiv c \mod{\, m \, n}$](img22.png) |
(1) |
where a convenient
is given by
![$\displaystyle c \ = \ a + (b - a) \, s \, m = b + (a - b) t\, n$](img24.png) |
(2) |
Therefore, for every
the system
of equations
![$\displaystyle \left\{ \begin{array}{l} x \equiv a \mod{\, m} \\ x \equiv b \mod{\, n} \\ \end{array} \right.$](img25.png) |
(3) |
has a solution.
Marc Moreno Maza
2008-01-31