Chinese Remainder Theorem - Finding The Solution With Basic Algebra and Modular Arithmetic

Finding The Solution With Basic Algebra and Modular Arithmetic

For example, consider the problem of finding an integer x such that

\begin{align} x &\equiv 2 \pmod{3}\\ x &\equiv 3 \pmod{4}\\ x &\equiv 1 \pmod{5}. \end{align}

A brute-force approach converts these congruences into sets and writes the elements out to the product of 3×4×5 = 60 (the solutions modulo 60 for each congruence):

x ∈ {2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53, 56, 59, …}
x ∈ {3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, …}
x ∈ {1, 6, 11, 16, 21, 26, 31, 36, 41, 46, 51, 56, …}.

To find an x that satisfies all three congruences, intersect the three sets to get:

x ∈ {11, …}.

Which can be expressed as

Another way to find a solution is with basic algebra, modular arithmetic, and stepwise substitution.

We start by translating these equivalences into equations for some t, s, and u:

  • Equation 1: x = 2 + 3 × t (mod 3)
  • Equation 2: x = 3 + 4 × s (mod 4)
  • Equation 3: x = 1 + 5 × u (mod 5).

Start by substituting the x from equation 1 into equivalence 2: 2 + 3 × t = 3 (mod 4), hence 3 × t = 1 (mod 4), or t = (1/3) (mod 4) = 3 (mod 4), meaning that t = 3 + 4 × s for integer s.

Plug t into equation 1: x = 2 + 3 × t (mod 3) = 2 + 3 × (3 + 4 × s) (mod 3) = 11 + 12 × s (mod 3).

Plug this x into equivalence 3: 11 + 12 × s = 1 (mod 5). Casting out 5s, we get 1 + 2 × s = 1 (mod 5), or 2 × s = 0 (mod 5), meaning that s = 0 + 5 × u for integer u.

Finally, x = 11 + 12 × s = 11 + 12 × (5 × u) = 11 + (60 × u). Since 60 = lcm(3, 4, 5), we have solutions 11, 71, 131, 191, …

Read more about this topic:  Chinese Remainder Theorem

Famous quotes containing the words finding the, finding, solution, basic, algebra and/or arithmetic:

    “What Romantic terminology called genius or talent or inspiration is nothing other than finding the right road empirically, following one’s nose, taking shortcuts.”
    —Italo Calvino (1923–1985)

    “Lais is now no lover of the glass,
    seeing no more the face as once it was,
    wishing to see that face and finding this.”
    —Hilda Doolittle (1886–1961)

    “I can’t quite define my aversion to asking questions of strangers. From snatches of family battles which I have heard drifting up from railway stations and street corners, I gather that there are a great many men who share my dislike for it, as well as an equal number of women who ... believe it to be the solution to most of this world’s problems.”
    —Robert Benchley (1889–1945)

    “... the basic experience of everyone is the experience of human limitation.”
    —Flannery O’Connor (1925–1964)

    “Poetry has become the higher algebra of metaphors.”
    —José Ortega Y Gasset (1883–1955)

    “Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build my arithmetic.... It is all the more serious since, with the loss of my rule V, not only the foundations of my arithmetic, but also the sole possible foundations of arithmetic seem to vanish.”
    —Gottlob Frege (1848–1925)