How do you solve Chinese remainder theorem problems?

Process to solve systems of congruences with the Chinese remainder theorem:

  1. Begin with the congruence with the largest modulus, x ≡ a k ( m o d n k ) .
  2. Substitute the expression for x x x into the congruence with the next largest modulus, x ≡ a k ( m o d n k ) ⟹ n k j k + a k ≡ a k − 1 ( m o d n k − 1 ) .

How is Chinese remainder calculated?

How to calculate Chinese remainder? To find a solution of the congruence system, take the numbers ^ni=nni=n1…ni−1ni+1… nk n ^ i = n n i = n 1 … n i − 1 n i + 1 … n k which are also coprimes. To find the modular inverses, use the Bezout theorem to find integers ui and vi such as uini+vi^ni=1 u i n i + v i n ^ i = 1 .

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.

How do you find the remainder theorem on a calculator?

The procedure to use the remainder theorem calculator is as follows:

  1. Step 1: Enter the numerator and denominator polynomial in the respective input field.
  2. Step 2: Now click the button “Divide” to get the output.
  3. Step 3: Finally, the quotient and remainder will be displayed in the new window.

What are the last two digits of 49 19 using Chinese remainder theorem?

Expert-verified answer The Chinese remainder theorem provides with a unique solution to simultaneous linear congruences with the coprime modulo. The modulo generally being 100. Hence, the last two digits of 49^19 is 49.

Why is it called 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.

What is mod in math?

Modulo is a math operation that finds the remainder when one integer is divided by another. In writing, it is frequently abbreviated as mod, or represented by the symbol %. For two integers a and b: a mod b = r. Where a is the dividend, b is the divisor (or modulus), and r is the remainder.

What is P C in remainder theorem?

Suppose p is a polynomial of degree at least 1 and c is a real number. When p(x) is divided by x−c the remainder is p(c).

How do you find the remainder when FX is divided by XK?

1 Expert Answer Remainder when f(x) is divided by x – k is f(k). So, remainder when f(x) is divided by x – (-3) is f(-3) = -81.

What are the last two digits of 7 to the power 2008?

as we can see it repeats at every 5th value. Hence last two digit in 7^2008 is 01.

How do you find the last two digits of a number?

Last two digits of a number is basically the tens place and units place digit of that number. So given a number say 1439, the last two digits of this number are 3 and 9, which is pretty straight forward.