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Mathematics

The Chinese Remainder Theorem and Modular Arithmetic

Quick fact

The Chinese Remainder Theorem guarantees that, given several remainders modulo pairwise coprime moduli, there is exactly one integer modulo the product of those moduli that satisfies all conditions at once—this is what allows RSA decryption to be performed several times faster than naive exponentiation.

Why this is interesting

Imagine you have a puzzle: a number leaves a remainder of 2 when divided by 3 and a remainder of 3 when divided by 5—but can you find it? The answer might surprise you: there isn’t just one, but an infinite family of solutions, and they all share a simple pattern.