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Posted 23 Sep 2008

Chinese Remainder Problem

, 17 Oct 2008
Solve the Chinese remainder problem cleverly
using System;
using System.Collections.Generic;
using System.Text;

namespace ChineseRemainderProblem
{
    /// <summary>
    /// Solve Chinese Remainder Problem
    /// </summary>
    public class RemainderProblem
    {
        /// <summary>
        /// Get Greatest Common Divisor
        /// </summary>
        public static int GCD(int a, int b)
        {
            if (a > b)
            {
                int t = b;
                b = a;
                a = t;
            }
            int r = b % a;
            while (r != 0)
            {
                b = a;
                a = r;
                r = b % a;
            }
            return a;
        }

        /// <summary>
        /// Get Least Common Multiple
        /// </summary>
        public static int LCM(int a, int b)
        {
            int gcd = GCD(a, b);
            return a * b / gcd;
        }

        /// <summary>
        /// Solve ax=by+1
        /// </summary>
        public static void Solve(int a, int b, out int x, out int y)
        {
            if (b == 1)
            {
                x = 1;
                y = a - 1;
            }
            else if (a == 1)
            {
                y = 1;
                x = b + 1;
            }
            else if (b > a)
            {
                int subx;
                Solve(a, b - a, out subx, out y);
                x = y + subx;
            }
            else if (a > b)
            {
                int suby;
                Solve(a - b, b, out x, out suby);
                y = x + suby;
            }
            else
            {
                throw new Exception(String.Format("The equation {0}x={1}y+1 has no integer solution.", a, b));
            }
        }

        /// <summary>
        /// Solve ax = by + c
        /// </summary>
        public static void Solve(int a, int b, int c, out int x1, out int y1)
        {
            /* if 
                   a * x0 = b * y0 + 1
               then
                   x = b * t + c * x0
                   y = a * t + c * y0
               satisfies
                   a * x  = b * y  + c
               so
                   x1 = (c * x0) mod b
                   y1 = (c * y0) mod a
             */

            int d = GCD(a, b);
            if (d > 1)
            {
                if (c % d != 0)
                {
                    throw new Exception(String.Format("The equation {0}x={1}y+{2} has no integer solution.", a, b, c));
                }
                a = a / d;
                b = b / d;
                c = c / d;
            }

            int x0, y0;
            Solve(a, b, out x0, out y0);

            x1 = (c * x0) % b;
            y1 = (c * y0) % a;
        }

        /// <summary>
        /// Solve a * x + r1 = b * y + r2
        /// </summary>
        public static void Solve(int a, int b, int r1, int r2, out int x1, out int y1)
        {
            if (r2 > r1)
            {
                Solve(a, b, r2 - r1, out x1, out y1);
            }
            else if (r1 > r2)
            {
                Solve(b, a, r1 - r2, out y1, out x1);
            }
            else
            {
                x1 = b;
                y1 = a;
            }
        }

        /// <summary>
        /// Solve a * x + r1 = b * y + r2 = c * z + r3 = n
        /// n0 = n mod t
        /// </summary>
        public static void Solve(int a, int b, int c, int r1, int r2, int r3, out int n0, out int t)
        {
            int x1, y1;
            Solve(a, b, r1, r2, out x1, out y1);

            // let r4 = a * x1 + r1 = b * y1 + r2
            // to satisfy n = a * x + r1 = b * y + r2
            // we can see n = a * b * u + r4

            int r4 = a * x1 + r1;

            int x2, y2;
            Solve(a * b, c, r4, r3, out x2, out y2);

            n0 = c * y2 + r3;
            t = LCM(LCM(a, b), c);
            //n0 = n % t;
        }
    }
}

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About the Author

Liu Junfeng
Architect YunCheDa Hangzhou
China China
No Biography provided

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