# Chinese Remainder Theorem

## Theorem

Let $a, b \in \Z$.

Let $r$ and $s$ be coprime integers.

Then:

$a \equiv b \pmod {r s}$ if and only if $a \equiv b \pmod r$ and $a \equiv b \pmod s$

where $a \equiv b \pmod r$ denotes that $a$ is congruent modulo $r$ to $b$.

### General Result

Let $b_1, b_2, \ldots, b_r \in \Z$.

Let $n_1, n_2, \ldots, n_r$ be pairwise coprime positive integers.

Let $\ds N = \prod_{i \mathop = 1}^r n_i$.

Then the system of linear congruences:

 $\ds x$ $\equiv$ $\ds b_1$ $\ds \pmod {n_1}$ $\ds x$ $\equiv$ $\ds b_2$ $\ds \pmod {n_2}$ $\ds$ $\vdots$ $\ds$ $\ds x$ $\equiv$ $\ds b_r$ $\ds \pmod {n_r}$

has a simultaneous solution which is unique modulo $N$.

### Ultrageneral Result

Let $n_1, n_2, \ldots, n_k$ be positive integers.

Let $b_1, b_2, \ldots, b_k$ be integers such that:

$\forall i \ne j: \gcd \set {n_i, n_j} \divides b_i - b_j$

Then the system of linear congruences:

 $\ds x$ $\equiv$ $\ds b_1$ $\ds \pmod {n_1}$ $\ds x$ $\equiv$ $\ds b_2$ $\ds \pmod {n_2}$ $\ds$ $\vdots$ $\ds$ $\ds x$ $\equiv$ $\ds b_k$ $\ds \pmod {n_k}$

has a simultaneous solution which is unique modulo $\lcm \set {n_1, \ldots, n_k}$.

## Proof

### Necessary Condition

This is proved in Congruence by Divisor of Modulus.

Note that for this result it is not required that $r \perp s$.

$\Box$

### Sufficient Condition

Now suppose that $a \equiv b \pmod r$ and $a \equiv b \pmod s$.

We have by definition of congruence:

$a \equiv b \pmod r \implies \exists k \in \Z: a - b = k r$

From $a \equiv b \pmod s$ we also have that:

$k r \equiv 0 \pmod s$

As $r \perp s$, we have from Common Factor Cancelling in Congruence:

$k \equiv 0 \pmod s$

So:

$\exists q \in \Z: a - b = q s r$

Hence by definition of congruence:

$a \equiv b \pmod {r s}$

$\blacksquare$

## Examples

### $n \equiv 7 \pmod {12}$ implies $n \equiv 3 \pmod 4$

Let $n \equiv 7 \pmod {12}$.

Then:

$n \equiv 3 \pmod 4$

## Warning

Let $r$ not be coprime to $s$.

Then it is not necessarily the case that:

$a \equiv b \pmod {r s}$ if and only if $a \equiv b \pmod r$ and $a \equiv b \pmod s$

where $a \equiv b \pmod r$ denotes that $a$ is congruent modulo $r$ to $b$.

## Historical Note

The Chinese Remainder Theorem was found in the Sun Tzu Suan Ching of Sun Tzu, who was active in China sometime between $200$ and $500$ C.E (nobody knows exactly when).

The principle is believed to have been used by Chinese calendar makers as far back as $100$ C.E.