The Chinese remainder theorem reconstructs a global residue class from compatible local residues. Algebraically, it decomposes a quotient by a product of coprime ideals into a product of simpler quotient rings.
Two moduli
Theorem. If \(\gcd(m,n)=1\), the map \[ \Phi:\mathbb Z/mn\mathbb Z\to \mathbb Z/m\mathbb Z\times\mathbb Z/n\mathbb Z,\qquad [x]_{mn}\mapsto([x]_m,[x]_n), \] is a ring isomorphism.
Proof. Choose \(r,s\in\mathbb Z\) with \(rm+sn=1\). For prescribed residues \(a\bmod m\), \(b\bmod n\), set \[ x=bsn+arm. \] Then \(x\equiv a\pmod m\) and \(x\equiv b\pmod n\), proving surjectivity. If \(x\) maps to zero, both \(m,n\mid x\); coprimality implies \(mn\mid x\), so the kernel is zero.
Several moduli and explicit idempotents
Corollary. For pairwise coprime \(n_1,\ldots,n_r\), every system \[ x\equiv a_i\pmod{n_i},\qquad1\le i\le r, \] has a unique solution modulo \(N=n_1\cdots n_r\).
Let \(N_i=N/n_i\) and choose \(u_iN_i\equiv1\pmod{n_i}\). Then \[ e_i=u_iN_i \] satisfies \(e_i\equiv1\pmod{n_i}\) and \(e_i\equiv0\pmod{n_j}\) for \(j\ne i\). The solution is \[ x\equiv\sum_{i=1}^ra_ie_i\pmod N. \] The elements \(e_i\) are orthogonal idempotents: \(e_i^2=e_i\), \(e_ie_j=0\), and \(\sum_ie_i=1\) modulo \(N\).
Example. Solve \[ x\equiv2\pmod3,\quad x\equiv3\pmod5,\quad x\equiv2\pmod7. \] Here \(N=105\). One may take \[ e_1=70,\qquad e_2=21,\qquad e_3=15. \] Thus \(x\equiv2(70)+3(21)+2(15)=233\equiv23\pmod{105}\).
Noncoprime moduli
Theorem. The pair \[ x\equiv a\pmod m,\qquad x\equiv b\pmod n \] is solvable if and only if \[ a\equiv b\pmod{\gcd(m,n)}. \] When solvable, the solution is unique modulo \(\operatorname{lcm}(m,n)\).
Proof. Subtracting the congruences makes the compatibility condition necessary. If it holds, write \(d=\gcd(m,n)\), \(b-a=dc\), and solve \((m/d)t\equiv c\pmod{n/d}\); the coefficient is invertible. Any two solutions differ by a common multiple of \(m,n\), hence by their lcm.
Prime-power reduction
Writing \(n=\prod_pp^{v_p(n)}\) gives \[ \mathbb Z/n\mathbb Z\cong \prod_{p\mid n}\mathbb Z/p^{v_p(n)}\mathbb Z. \] Consequently a congruence modulo \(n\) can be solved independently at each prime power and then recombined. This local-to-global pattern reappears in \(p\)-adic analysis, algebraic number fields, and adelic methods.