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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.

Lemma

Let \(m\) and \(n\) be positive integers such that \(\gcd( m, n) = 1\). Then for \(a, b \in {\mathbb Z}\) the system \[\begin{aligned}x & \equiv a \pmod{m} \\ x & \equiv b \pmod{n}\end{aligned}\] has a solution. If \(x_1\) and \(x_2\) are two solutions of the system, then \(x_1 \equiv x_2 \pmod{mn}\).

Proof

The equation \(x \equiv a \pmod{m}\) has a solution since \(a + km\) satisfies the equation for all \(k \in {\mathbb Z}\). We must show that there exists an integer \(k_1\) such that \[\begin{aligned}\end{aligned}\]. This is equivalent to showing that \[\begin{aligned}\end{aligned}\] has a solution for \(k_1\). Since \(m\) and \(n\) are relatively prime, there exist integers \(s\) and \(t\) such that \(ms + nt = 1\). Consequently, \[\begin{aligned}\end{aligned}\], or \[\begin{aligned}\end{aligned}\]. Now let \(k_1 = (b-a)s\).

To show that any two solutions are congruent modulo \(mn\), let \(c_1\) and \(c_2\) be two solutions of the system. That is, \[\begin{aligned}c_i & \equiv a \pmod{m} \\ c_i & \equiv b \pmod{n}\end{aligned}\] for \(i = 1, 2\). Then \[\begin{aligned}c_2 & \equiv c_1 \pmod{m} \\ c_2 & \equiv c_1 \pmod{n}\end{aligned}\]. Therefore, both \(m\) and \(n\) divide \(c_1 - c_2\). Consequently, \(c_2 \equiv c_1 \pmod{mn}\).

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.

Condensed: the full section is in Judson, Abstract Algebra: Theory and Applications.

Ugbu a ị Enweghị kalkulata na-ewepụ nke a, ma ọ bụ na ọ nwere ike ijikwa ya. Jiri otu n'okpuru, mọọbụ pịa nke gị.

Kwụsị ọrụ gị

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Akaụntụ Nbanye

Aịkọn ndị ahụ ejirila ebe a

Tinye ọbụla simbụl maka nkọwa zuru ezu, inyogo, na ihe ọbụla ọbụla na-egosi.

Ajụjụ ndị mmadụ jụrụ

What is a group, in plain words?

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.

Oge nke ihuakwụkwọ a bụ nke a gbanwere site na Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). E kọwaara ya na ebe a; nsogbu bụ nke anyị.

Oge Abstract Algebra

Groups Subgroups, cosets and Lagrange's theorem Cyclic groups and permutation groups Homomorphisms, normal subgroups and quotient groups Rings and fields Galois theory: why the quintic has no formula