Formulas.

Let m and n be two relatively prime elements of an Euclidean domain R. (You may think R = $\mbox{${\mathbb Z}$}$ or R = $\mbox{${\mathbb Q}$}$[x].) Let s, t $\in$ R be such that s m + t n = 1. For every a, b $\in$ R there exists c $\in$ R such that

$$\forall \in \left\{\vphantom{ \begin{array}{l} x \equiv a \mod{\, m} \\ x \equiv b \mod{\, n} \\ \end{array} }\right. \begin{array}{l} x \equiv a \mod{\, m} \\ x \equiv b \mod{\, n} \\ \end{array} \iff \equiv$$ (1)

where a convenient c is given by

c  =  a + (b - a) s m = b + (a - b)t n (2)

Therefore, for every a, b $\in$ R the system of equations

$$\left\{\vphantom{ \begin{array}{l} x \equiv a \mod{\, m} \\ x \equiv b \mod{\, n} \\ \end{array} }\right. \begin{array}{l} x \equiv a \mod{\, m} \\ x \equiv b \mod{\, n} \\ \end{array}$$ (3)

has a solution. We give also

$${\frac{{1}}{{4}}} {\frac{{1}}{{16}}} {\frac{{1}}{{2^{2k}}}} {\frac{{1}}{{2}}} {\frac{{1}}{{4}}} {\frac{{1}}{{8}}} {\frac{{1}}{{16}}} {\frac{{1}}{{2^{2k}}}}$$ (4)