[PDF] chinese remainder theorem proof pdf

  • How do you prove the Chinese remainder theorem?

    Suppose that x and y are both solutions to all the congruences.
    As x and y give the same remainder, when divided by ni, their difference x ? y is a multiple of each ni.
    As the ni are pairwise coprime, their product N also divides x ? y, and thus x and y are congruent modulo N.

  • What is an example of the Chinese remainder theorem with a solution?

    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.

  • What is Chinese remainder theorem powers?

    The Chinese remainder theorem is a powerful tool to find the last few digits of a power.
    The idea is to find a number mod 5 n 5^n 5n and mod 2 n , 2^n, 2n, and then combine those results, using the Chinese remainder theorem, to find that number mod 1 0 n 10^n 10n.

  • What is Chinese remainder theorem powers?

    Description. mod( x , div ) = res .
    In other words, dividing x by each element of div leaves as remainder the corresponding element of res .

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