Задача 13.

* (Chinese remainder theorem) Let numbers $a_1$, $a_2$, ..., $a_n$ be pairwise co-prime (i.e.the only positive integer that divides both of them is 1). Then for any $x_1$, $x_2$, ..., $x_n$ there exists $x$ such that \begin{equation*} x\equiv x_i\pmod {a_i}   i=1,\ldots,n, \end{equation*} moreover, any two numbers, which satisfy this property, are congruent modulo $a_1 a_2 \ldots a_n$.