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:

    “It makes little sense to spend a month teaching decimal fractions to fourth-grade pupils when they can be taught in a week, and better understood and retained, by sixth-grade students. Child-centeredness does not mean lack of rigor or standards; it does mean finding the best match between curricula and children’s developing interests and abilities.”
    —David Elkind (20th century)

    “One of the many reasons for the bewildering and tragic character of human existence is the fact that social organization is at once necessary and fatal. Men are forever creating such organizations for their own convenience and forever finding themselves the victims of their home-made monsters.”
    —Aldous Huxley (1894–1963)

    “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 tool for the manipulation of reality is the manipulation of words. If you can control the meaning of words, you can control the people who must use the words.”
    —Philip K. Dick (1928–1982)

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

    “O! O! another stroke! that makes the third.
    He stabs me to the heart against my wish.
    If that be so, thy state of health is poor;
    But thine arithmetic is quite correct.”
    —A.E. (Alfred Edward)