Finding The Solution With Basic Algebra and Modular Arithmetic
For example, consider the problem of finding an integer x such that
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:
“As a father I had some trouble finding the words to separate the person from the deed. Usually, when one of my sons broke the rules or a window, I was too angry to speak calmly and objectively. My own solution was to express my feelings, but in an exaggerated, humorous way: “You do that again and you will be grounded so long they will call you Rip Van Winkle II,” or “If I hear that word again, I’m going to braid your tongue.””
—David Elkind (20th century)
“As one delves deeper and deeper into Etiquette, disquieting thoughts come. That old Is- It-Worth-It Blues starts up again softly, perhaps, but plainly. Those who have mastered etiquette, who are entirely, impeccably right, would seem to arrive at a point of exquisite dullness. The letters and the conversations of the correct, as quoted by Mrs. Post, seem scarcely worth the striving for. The rules for finding topics of conversation fall damply on the spirit.”
—Dorothy Parker (1893–1967)
“I herewith commission you to carry out all preparations with regard to ... a total solution of the Jewish question in those territories of Europe which are under German influence.... I furthermore charge you to submit to me as soon as possible a draft showing the ... measures already taken for the execution of the intended final solution of the Jewish question.”
—Hermann Goering (1893–1946)
“It is a strange fact that freedom and equality, the two basic ideas of democracy, are to some extent contradictory. Logically considered, freedom and equality are mutually exclusive, just as society and the individual are mutually exclusive.”
—Thomas Mann (1875–1955)
“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)