When was the Chinese remainder theorem?
Chinese remainder theorem, ancient theorem that gives the conditions necessary for multiple equations to have a simultaneous integer solution. The theorem has its origin in the work of the 3rd-century-ad Chinese mathematician Sun Zi, although the complete theorem was first given in 1247 by Qin Jiushao.
How do you implement Chinese remainder theorem?
How to implement the Chinese Remainder Theorem in Java
- What do we need to find?
- Step 1: Find the product of all the numbers in the first array.
- Step 2: Find the partial product of each number.
- Find the modular multiplicative inverse of number[i] modulo partialProduct[i].
- Step 4: Final Sum.
- Step 5: Return the smallest X.
What is Chinese remainder theorem give an example?
In the above example, the product is 3*4*5 = 60. And 11 is one solution, other solutions are 71, 131, .. etc. All these solutions produce the same remainder when divided by 60, i.e., they are of form 11 + m*60 where m >= 0.
What is Chinese remainder theorem example?
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 the importance of remainder theorem?
The remainder theorem enables us to calculate the remainder of the division of any polynomial by a linear polynomial, without actually carrying out the steps of the division algorithm. When a polynomial a(x) is divided by a linear polynomial b(x) whose zero is x = k, the remainder is given by r = a(k).
How does remainder theorem work?
The remainder theorem states that when a polynomial, f(x), is divided by a linear polynomial , x – a, the remainder of that division will be equivalent to f(a). It should be noted that the remainder theorem only works when a function is divided by a linear polynomial, which is of the form x + number or x – number.