The list of moduli m must be pairwise relatively prime positive integers. (For the case of non-coprime moduli, see NumberTheory[ChineseRemainder].) Both lists u and m must be the same length $n$. The list of images u need not be reduced modulo m on input. In the following, $M$ denotes the product of the moduli.

If u is a list of integers, chrem(u, m) computes the unique positive integer a such that $a\mathbf{mod}\mathrm{m1}=\mathrm{u1},a\mathbf{mod}\mathrm{m2}=\mathrm{u2},...,a\mathbf{mod}\mathrm{mn}=\mathrm{un}$ , and $0\le a<M$.

If the global variable mod has been assigned to mods then the result $a$ is returned in the symmetric range for the integers modulo $M$. For example, the symmetric range for the integers modulo $M=35$ is $-17\le a\le \mathrm{+17}$.

If u is a list of polynomials, chrem is applied across the polynomials so that the output $f$ is a polynomial satisfying $f\mathbf{mod}\mathrm{m1}=\mathrm{u1}$ , ..., $f\mathbf{mod}\mathrm{mn}=\mathrm{un}$.

If u is a list of lists, chrem is applied across the lists so that the output will be a list $L$ satisfying $L\mathbf{mod}\mathrm{m1}=\mathrm{u1}$, ..., $L\mathbf{mod}\mathrm{mn}=\mathrm{un}$ .

For a definition, see Chinese remainder theorem.

$\mathrm{chrem}\left(\left[1\&comma;2\right]\&comma;\left[5\&comma;7\right]\right)$

$16$

$\mathrm{chrem}\left(\left[3x+1\&comma;x+2y+2\right]\&comma;\left[5\&comma;7\right]\right)$

$8x+16+30y$

$\mathrm{chrem}\left(\left[\left[3\&comma;0\&comma;1\right]\&comma;\left[1\&comma;2\&comma;2\right]\right]\&comma;\left[5\&comma;7\right]\right)$

$\left[8\&comma;30\&comma;16\right]$

$\mathrm{`mod`}≔\mathrm{mods}$

$\mathrm{mod}≔\mathrm{mods}$

$\mathrm{chrem}\left(\left[3x+1\&comma;x+2y+2\right]\&comma;\left[5\&comma;7\right]\right)$

$8x+16-5y$