# Proof: By Induction

(related to Theorem: Chinese Remainder Theorem)

By hypothesis, $m_1,m_2,\ldots,m_r$ are positive integers and $a_1,\ldots,a_r$ are any integers. We will first prove the theorem for $r=2$ and then the general case by induction. We can assume all $a_i$ are distinct, otherwise, we could replace two congruences of the system by just one and reduce the number of simultaneous congruences. For the case $r=1,$ the Chinese remainder theorem is trivial.

### Base Case $r=2$

We will first show that both congruences are solvable simultaneously if and only if the greatest common divisor $d:=\gcd(m_1,m_2)$ is a divisor of $a_1-a_2.$

### "$\Rightarrow$"

• Assume, $x(m_1)\equiv a_1(m_1)$ and $x(m_2)\equiv a_1(m_2).$
• From the congruence modulo a divisor, it follows that $x(d)\equiv a_1(d)$ and $x(d)\equiv a_1(d).$
• Thus, $a_1(d)=a_2(d).$
• By definition of congruence, it follows $d\mid (a_1-a_2).$

### "$\Leftarrow$"

• Assume, $d\mid (a_1-a_2).$
• It follows, $d\mid (a_2-a_1).$
• According to existence and number of solutions of an LDE with one variable, the congruence $(m_1y)(m_2)\equiv (a_2-a_1)(m_2)\label{eq:E18592a}\tag{1}$ is solvable if and only if $\gcd(m_1,m_2)=d\mid (a_2-a_1),$ and it has exactly $d$ solutions.
• By setting $x:=m_1y$ we have $(x+a_1)(m_2)\equiv a_2(m_2).$
• On the other side, the congruence class $x(m_1)\equiv a_1(m_1)\label{eq:E18592b}\tag{2}$ is equivalent to $x=a_1+m_1h$ for all $h\in\mathbb Z.$
• Now we can chose $h$ such that $x(m_2)\equiv(hm_1+a_1)(m_2)\equiv a_2(m_2).$
• Comparing both results, we have $y=h$ and $x(m_2)\equiv(x+a_1)(m_2)\equiv a_2(m_2).\label{eq:E18592c}\tag{3}$
• It follows from $(\ref{eq:E18592b})$ and $(\ref{eq:E18592c})$ that $x(m_1)\equiv a_1(m_1)$ and $x(m_2)\equiv a_1(m_2).$

Now, we will show that all simultaneous solutions belong to a single congruence class modulo the least common multiple $\operatorname{lcm}(m_1,m_2).$

• In particular, the congruence $(\ref{eq:E18592a})$ is solvable for an $y_0$ with $$\frac{m_1}{d}y_0\equiv\frac{a_2-a_1}{d}\mod \frac{m_2}{d},$$ which has exactly one solution according to the existence and number of solutions of an LDE with one variable.
• All $x$ solving the simultaneous congruences $x(m_1)\equiv a_1(m_1)$ and $x(m_2)\equiv a_1(m_2)$ can be written (for $h\in\mathbb Z$) as $$\begin{array}{rcl}x&=&a_1+\left(y_0+h\frac{m_2}{d}\right)m_1\\ &=&a_1+y_0m_1+h\frac{m_1m_2}{d}\\ &=&a_1+y_0m_1+h\operatorname{lcm}(m_1,m_2), \end{array}$$ because of the relationship between the greatest vommon divisor and the least common multiple.
• For $h\in\mathbb Z,$ the last equation consists exactly of the congruence class modulo $\operatorname{lcm}(m_1,m_2).$

### Induction step $r > 2.$

• Let $r > 2$ and let the claim be proven for $r-1.$
• This means that there is an $a\in\mathbb Z$ with $x\equiv a\mod \operatorname{lcm}(m_1,\ldots,m_{r-1})$ being a solution of the simultaneous $r-1$ congruence classes $$\begin{array}{rcl} x(m_1)&\equiv&a_1(m_1)\\ x(m_2)&\equiv&a_2(m_2)\\ &\vdots&\\ x(m_{r-1})&\equiv& a_{r-1}(m_{r-1}).\\ \end{array}\label{eq:E18598d}\tag{4}$$
• The step $r-1\to r$ follows from the base case applied to the congruence system $$\begin{array}{rcl} x(\operatorname{lcm}(m_1,\ldots,m_{r-1}))&\equiv&a(\operatorname{lcm}(m_1,\ldots,m_{r-1}))\\ x(m_r)&\equiv&a_r(m_r),\\ \end{array}$$ and the definition of the least common multiple with more than two variables $\operatorname{lcm}(m_1,\ldots,m_{r})=\operatorname{lcm}(\operatorname{lcm}(m_1,\ldots,m_{r-1}), m_r).$

Github: ### References

#### Bibliography

1. Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927