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)
“In the middle years of childhood, it is more important to keep alive and glowing the interest in finding out and to support this interest with skills and techniques related to the process of finding out than to specify any particular piece of subject matter as inviolate.”
—Dorothy H. Cohen (20th century)
“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)
“The basic Female body comes with the following accessories: garter belt, panti-girdle, crinoline, camisole, bustle, brassiere, stomacher, chemise, virgin zone, spike heels, nose ring, veil, kid gloves, fishnet stockings, fichu, bandeau, Merry Widow, weepers, chokers, barrettes, bangles, beads, lorgnette, feather boa, basic black, compact, Lycra stretch one-piece with modesty panel, designer peignoir, flannel nightie, lace teddy, bed, head.”
—Margaret Atwood (b. 1939)
“Poetry has become the higher algebra of metaphors.”
—José Ortega Y Gasset (1883–1955)
“Under the dominion of an idea, which possesses the minds of multitudes, as civil freedom, or the religious sentiment, the power of persons are no longer subjects of calculation. A nation of men unanimously bent on freedom, or conquest, can easily confound the arithmetic of statists, and achieve extravagant actions, out of all proportion to their means; as, the Greeks, the Saracens, the Swiss, the Americans, and the French have done.”
—Ralph Waldo Emerson (1803–1882)