Update March 16, 2018
Chinese Puzzle: The Intersection Algorithm

What we needed is a method of solving the following problem: Given a set of remaiders, what is the number that would produce these remaiders.

Example What is the number x ( in the range 1-15) such that x%3=1, x%5=3? Sometimes you can see notation x mod 3 = 1, x mod 5 = 3. The meaning of these equations is that x - 1 is divisible by 3, and x - 3 is divisible by 5. The answer is x=13 because 13 = 4x3+1 and 13 = 2x5 + 3. This solution is unique in the range 1 to 15.

So what are the methods of solution? There is more then one method and we highlight here just one that we call the intersection method.

Using the above example we can split the problem into two problems. The problem 1 is to find x such that x%3 = 1 The problem 2 is to find x such that x%5 = 3 The solution set for problem 1 is S1 = {1,4,7,10,13}. The solution set for problem 2 is S2 = {3,8,13}. The elements of the intersection S1 and S2 satisfy both equations. The intesection contains the only one element 13 and that is the solution , the number x=13. This example illustrates the general intersection algorithm. The intersection algorithm can be stated as : Compute solution sets for individual divisors and remainders. The solution is the unique element of the individual solution sets. QED

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