What is the Chinese remainder theorem explain with example?

In number theory, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime.

Why is it called 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.

Why is it called Chinese remainder theorem?

What are simultaneous congruences?

Solution. A solution of a system of simultaneous congruences is a residue class modulo lcm{n1,n2,…,nr} such that any element of that class satisfies all the congruences.

How do you simplify congruences?

To solve a linear congruence ax ≡ b (mod N), you can multiply by the inverse of a if gcd(a,N) = 1; otherwise, more care is needed, and there will either be no solutions or several (exactly gcd(a,N) total) solutions for x mod N.

How to solve a system of congruence with the Chinese Remainder Theorem?

Process to solve systems of congruences with the Chinese remainder theorem: For a system of congruences with co-prime moduli, the process is as follows: x ≡ a k ( m o d n k). ). Re-write this modulus as an equation, j k. . 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 the Chinese Remainder Theorem used in RSA?

On this page we look at the Chinese Remainder Theorem (CRT), Gauss’s algorithm to solve simultaneous linear congruences, a simpler method to solve congruences for small moduli, and an application of the theorem to break the RSA algorithm when someone sends the same encrypted message to three different recipients using the same exponent of e=3.

How to find solution to system of congruences?

The following is a general construction to find a solution to a system of congruences using the Chinese remainder theorem: . y i = N n i = n 1 n 2 ⋯ n i − 1 n i + 1 ⋯ n k. . are pairwise coprime). . v v to the system of congruences. Then u ≡ v ( m o d n 1 n 2 ⋯ n k). ). .

How many eggs are left over in the Chinese Remainder Theorem?

Brahmagupta has a basket full of eggs. When he takes the eggs out of the basket 2 at a time, there is 1 egg left over. When he takes them out 3 at a time, there are 2 eggs left over. Likewise, when he takes the eggs out 4, 5, and 6 at a time, he finds remainders of 3, 4, and 5, respectively.