How do you calculate Chinese Remainder Theorem?

How do you calculate Chinese Remainder Theorem?

The Chinese Remainder Theorem

1. x = 2 ( mod 5 )
2. x = 3 ( mod 7 )
3. x = a ( mod p )
4. x = b ( mod q )
5. y = a q q 1 + b p p 1 ( mod p q )

What is the remainder theorem formula?

The remainder theorem formula is: p(x) = (x-c)·q(x) + r(x). The basic formula to check the division is: Dividend = (Divisor × Quotient) + Remainder.

How do you use the Chinese Remainder Theorem examples?

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 Chinese Remainder Theorem called in China?

As this theorem appears in a book in ancient China called Sunzi suanjing, it is called the “Chinese remainder theorem” in the West, and it deals with the remainder from the division of one number by another number.

Is Chinese remainder theorem if and only if?

The Chinese remainder theorem (CRT) asserts that there is a unique class a + NZ so that x solves the system (2) if and only if x ∈ a + NZ, i.e. x ≡ a(mod N). Thus the system (2) is equivalent to a single congruence modulo N.

How do you find the remainder theorem?

Verification: Given, the divisor is (x + 1), i.e. it is a factor of the given polynomial p(x). Remainder = Value of p(x) at x = -1. Hence proved the remainder theorem.

Why is it called the Chinese remainder theorem?

There was a much greater emphasis on algorithms used to solve more or less practical problems. The Chinese remainder theorem in its original form was an algorithm devised to solve a problem often known as “an unknown quantity of things” (物不知數), so that was the name used to refer to the original method.

What is the remainder when the polynomial?

This illustrates the Remainder Theorem. If a polynomial f(x) is divided by x−a , the remainder is the constant f(a) , and f(x)=q(x)⋅(x−a)+f(a) , where q(x) is a polynomial with degree one less than the degree of f(x) . Synthetic division is a simpler process for dividing a polynomial by a binomial.

Why is it called Chinese remainder theorem?

What is the history of the name “Chinese Remainder Theorem”?

Some time in the first century AD a Chinese mathematician by the name of Sun Zi published a book, Sun Zi Suanjing, or “The Arithmetical Classic of Sun Zi”. In this book Sun Zi introduced a method of solving systems of linear congruences that became known as the Chinese Remainder Theorem.

Who invented the Chineses 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 . The Chinese remainder theorem addresses the following type of problem.

How do you calculate a remainder?

How to calculate the remainder Begin with writing down your problem. Decide on which of the numbers is the dividend, and which is the divisor. Perform the division – you can use any calculator you want. Round this number down. Multiply the number you obtained in the previous step by the divisor.

What is synthetic division and remainder theorem?

Remainder Theorem. To find the remainder of a polynomial divided by some linear factor, we usually use the method of Polynomial Long Division or Synthetic Division. However, the concept of the Remainder Theorem provides us with a straightforward way to calculate the remainder without going into the hassle.