[PDF] chinese remainder theorem to solve congruences

The Chinese remainder theorem says we can uniquely solve every pair of congruences having relatively prime moduli. x ? a mod m, x ? b mod n has a solution, and this solution is uniquely determined modulo mn. What is important here is that m and n are relatively prime. There are no constraints at all on a and b.
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  • What is the Chinese remainder theorem congruence?

    The Chinese remainder theorem asserts that if the ni are pairwise coprime, and if a1, , ak are integers such that 0 ? ai < ni for every i, then there is one and only one integer x, such that 0 ? x < N and the remainder of the Euclidean division of x by ni is ai for every i.

  • What is the Chinese remainder theorem used for?

    The Chinese remainder theorem is frequently used when working with large integers because it enables the replacement of a computation for which one knows a bound on the size of the output by a number of smaller computations.

  • What is the Chinese remainder theorem used for?

    Example: Solve the simultaneous congruences x ? 6 (mod 11), x ? 13 (mod 16), x ? 9 (mod 21), x ? 19 (mod 25).
    Solution: Since 11, 16, 21, and 25 are pairwise relatively prime, the Chinese Remainder Theorem tells us that there is a unique solution modulo m, where m = 11?16?21?25 = 92400.

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