Chinese Remainder theorem with non-pairwise coprime moduli

Let $n_1,…,n_k \in \mathbb{N}$ and let $a_1,…,a_k \in \mathbb{Z}$.
How to prove the following version of the Chinese remainder theorem (see here):

There exists a $x \in \mathbb{Z}$ satisfying system of equations:
$$x=a_1 \pmod {n_1}$$
$$x=a_2 \pmod {n_2}$$
$$\ldots$$
$$x=a_k \pmod{n_k}$$
if and only if $a_i=a_j \pmod{\gcd(n_i,n_j)}$ for all $i,j=1,…,k$?

If numbers $n_i$, for $i=1,…,k$, are pairwise coprime, it is a classical version of Chinese remainder theorem.

Thanks.

Answer

If we factor $n_k$ into primes, $n_k = p_{1}^{b_{1}}\cdots p_r^{b_{r}}$, then the Chinese Remainder Theorem tells us that $x\equiv a_k\pmod{n_k}$ is equivalent to the system of congruences
$$\begin{align*}
x&\equiv a_k\pmod{p_1^{b_{1}}}\\
x&\equiv a_k\pmod{p_2^{b_{2}}}\\
&\vdots\\
x&\equiv a_k\pmod{p_r^{b_{r}}}
\end{align*}$$
Thus, we can replace the given system of congruences with one in which every modulus is a prime power, $n_i = p_i^{b_i}$.

Note that the assumption that $a_i\equiv a_j\pmod{\gcd(n_i,n_j)}$ “goes through” this replacement (if they were congruend modulo $\gcd(n_i,n_j)$, then they are congruent modulo the gcds of the prime powers as well).

So, we may assume without loss of generality that every modulus is a prime power.

I claim that we can deal with each prime separately, again by the Chinese Remainder Theorem. If we can solve all congruences involving the prime $p_1$ to obtain a solution $x_1$ (which will be determined modulo the highest power of $p_1$ that occurs); and all congruences involving the prime $p_2$ to obtain a solution $x_2$ (which will be determined modulo the highest power of $p_2$ that occurs); and so on until we obtain a solution $x_n$ for all congruences involving the prime $p_n$ (determined modulo the highest power of $p_n$ that occurs), then we can obtain a simultaneous solution by solving the usual Chinese Remainder Theorem system
$$\begin{align*}
x &\equiv x_1 \pmod{p_1^{m_1}}\\
&\vdots\\
x &\equiv x_n\pmod{p_n^{m_n}}
\end{align*}$$
(where $m_i$ is the highest power of $p_i$ that occurs as a modulus).

So we are reduced to solving figuring out whether we can solve the system
$$\begin{align*}
x &\equiv a_1\pmod{p^{b_1}}\\
x &\equiv a_2\pmod{p^{b_2}}\\
&\vdots\\
x & \equiv a_n\pmod{p^{b_n}}
\end{align*}$$
with, without loss of generality, $b_1\leq b_2\leq\cdots\leq b_n$.

When can this be solved? Clearly, this can be solved if and only if $a_i\equiv a_j\pmod{p^{b_{\min(i,j)}}}$: any solution must satisfy this condition, and if this condition is satisfied, then $a_n$ is a solution.

For example: say the original moduli had been $n_1 = 2^3\times 3\times 7^2$, $n_2= 2^2\times 5\times 7$, $n_3=3^2\times 5^3$. First we replace the system with the system of congruences
$$\begin{align*}
x&\equiv a_1 \pmod{2^3}\\
x&\equiv a_2\pmod{2^2}\\
x&\equiv a_1\pmod{3}\\
x&\equiv a_3\pmod{3^2}\\
x&\equiv a_2\pmod{5}\\
x&\equiv a_3\pmod{5^3}\\
x&\equiv a_1\pmod{7^2}\\
x&\equiv a_2\pmod{7}.
\end{align*}$$
Then we separately solve the systems:
$$\begin{align*}
x_1&\equiv a_1 \pmod{2^3} &x_2&\equiv a_1\pmod{3}\\
x_1&\equiv a_2\pmod{2^2}&x_2&\equiv a_3\pmod{3^2}\\
\strut\\
x_3&\equiv a_2\pmod{5}&x_4&\equiv a_1\pmod{7^2}\\
x_3&\equiv a_3\pmod{5^3}&x_4&\equiv a_2\pmod{7}.
\end{align*}$$

Assuming we can solve these, $x_1$ is determined modulo $2^3$, $x_2$ modulo $3^2$, $x_3$ modulo $5^3$, and $x_4$ modulo $7^2$, so we then solve the system
$$\begin{align*}
x &\equiv x_1\pmod{2^3}\\
x &\equiv x_2\pmod{3^2}\\
x&\equiv x_3 \pmod{5^3}\\
x&\equiv x_4\pmod{7^2}
\end{align*}$$
and obtain a solution to the original system.

Hence, if the condition $a_i\equiv a_j\pmod{\gcd(n_i,n_j)}$ holds in the original system, then we obtain a solution for each prime, and from the solution for each prime we obtain a solution to the original system by applying the usual Chinese Remainder Theorem twice.

Attribution
Source : Link , Question Author : Richard , Answer Author : Arturo Magidin

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