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:

    “Finding the perfect balance is getting harder and harder. We need to teach our children to be cautious without imparting fear, to learn right from wrong without being judgmental, to be assertive but not pushy, to stick to routines without sacrificing spontaneity, and to be determined but not stubborn.”
    —Fred G. Gosman (20th century)

    “The lion cares less about being king of the beasts than about finding his dinner.”
    —Mason Cooley (b. 1927)

    “There’s one solution that ends all life’s problems.”
    —Chinese proverb.

    “It is easier to move rivers and mountains than to change a person’s basic nature.”
    —Chinese proverb.

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

    “‘Tis no extravagant arithmetic to say, that for every ten jokes,—thou hast got an hundred enemies; and till thou hast gone on, and raised a swarm of wasps about thine ears, and art half stung to death by them, thou wilt never be convinced it is so.”
    —Laurence Sterne (1713–1768)