Why it matters

CRT is foundational for modular systems + RSA + big-number multiplication. Understanding shapes technique.

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The architecture

Combine two congruences at a time.

Use extended Euclidean.

CRT combinationCongruence 1x = a1 mod m1Congruence 2x = a2 mod m2Combineextended EuclidGeneral CRT allows non-coprime moduli with compatibility check
CRT algorithm.
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How it works end to end

Extended Euclidean: find g = gcd, u, v with u*m1 + v*m2 = g.

Combined solution: mod (m1*m2/g).

Non-coprime: check a1 = a2 mod g.