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:
“Love has its own instinct, finding the way to the heart, as the feeblest insect finds the way to its flower, with a will which nothing can dismay nor turn aside.”
—Honoré De Balzac (1799–1850)
“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)
“To the questions of the officiously meddling police Falter replied absently and tersely; but, when he finally grew tired of this pestering, he pointed out that, having accidentally solved “the riddle of the universe,” he had yielded to artful exhortation and shared that solution with his inquisitive interlocutor, whereupon the latter had died of astonishment.”
—Vladimir Nabokov (1899–1977)
“There’s one basic rule you should remember about development charts that will save you countless hours of worry.... The fact that a child passes through a particular developmental stage is always more important than the age of that child when he or she does it. In the long run, it really doesn’t matter whether you learn to walk at ten months or fifteen months—as long as you learn how to walk.”
—Lawrence Kutner (20th century)
“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)