# Documentation

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# `numlib`::`ichrem`

Chinese remainder theorem for integers

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## Syntax

```numlib::ichrem(`a`, `m`)
```

## Description

`numlib::ichrem(a,m)` returns the least nonnegative integer x such that for i = 1, …, nops(m) if such a number exists; otherwise `numlib::ichrem(a,m)` returns `FAIL`.

The entries in `m` need not be pairwise coprime.

`numlib::ichrem(a,m)` returns an error if `a` is not a list of integers or `m` is not a list of natural numbers or `a` and `m` are not lists of the same length.

## Examples

### Example 1

Here the moduli are pairwise coprime. In this case, a solution always exists:

`numlib::ichrem([2,3,2],[3,5,7])`

### Example 2

Here the moduli are not pairwise coprime, and a solution does not exist:

`numlib::ichrem([5,6,8],[20,21,22])`

### Example 3

Also here the moduli are not pairwise coprime, but a solution nevertheless exists:

`numlib::ichrem([5,6,7],[20,21,22])`

## Parameters

 `a` A list of integers `m` A list of natural numbers of the same length as `a`

## Return Values

Either a nonnegative integer or `FAIL`.