The Chinese Remainder Theorem is a result from elementary number theory about the solution of systems of simultaneous congruences. The Chinese mathematician Sun-tsï wrote about the theorem in the first century A.D.
An Application to Software Design
The Chinese Remainder Theorem is a result from elementary number theory about the solution of systems of simultaneous congruences.
The Chinese mathematician Sun-tsï wrote about the theorem in the first century A.D. This theorem has some interesting consequences in the design of software for parallel processors.
Example
Let us solve the system
\[\begin{aligned}x & \equiv 3 \pmod{4} \\ x & \equiv 4 \pmod{5}\end{aligned}\].
Using the Euclidean algorithm,
we can find integers \(s\) and \(t\) such that \(4s + 5t = 1\).
Two such integers are \(s = 4\) and \(t = -3\).
Consequently,
\[\begin{aligned}\end{aligned}\].
Special software is required for calculations involving larger integers which cannot be added directly by the machine.
By using the Chinese Remainder Theorem we can break down large integer additions and multiplications into calculations that the computer can handle directly.
This is especially useful on parallel processing computers which have the ability to run several programs concurrently.
A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group. That is where the idea came from.
What is the difference between a ring and a field?
A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.