The notion of congruences was first introduced and used by Carl Friedrich Gauss in his Disquisitiones Arithmeticae of 1801. The Chinese remainder theorem appears in Gauss's 1801 book Disquisitiones Arithmeticae. The result was later generalized with a complete solution called Da-yan-shu ( 大衍術) in Qin Jiushao's 1247 Mathematical Treatise in Nine Sections which was translated into English in early 19th century by British missionary Alexander Wylie. Special cases of the Chinese remainder theorem were also known to Brahmagupta (7th century), and appear in Fibonacci's Liber Abaci (1202). What amounts to an algorithm for solving this problem was described by Aryabhata (6th century). Sunzi's work contains neither a proof nor a full algorithm. If we count them by threes, we have two left over by fives, we have three left over and by sevens, two are left over. There are certain things whose number is unknown. The earliest known statement of the theorem, as a problem with specific numbers, appears in the 3rd-century book Sunzi Suanjing by the Chinese mathematician Sunzi: It has been generalized to any ring, with a formulation involving two-sided ideals. The Chinese remainder theorem (expressed in terms of congruences) is true over every principal ideal domain. The Chinese remainder theorem is widely used for computing with large integers, as it allows replacing a computation for which one knows a bound on the size of the result by several similar computations on small integers. The earliest known statement of the theorem is by the Chinese mathematician Sunzi in the Sunzi Suanjing in the 3rd century CE. Importantly, this tells us that if n is a natural number less than 105, then 23 is the only possible value of n. In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1).įor example, if we know that the remainder of n divided by 3 is 2, the remainder of n divided by 5 is 3, and the remainder of n divided by 7 is 2, then without knowing the value of n, we can determine that the remainder of n divided by 105 (the product of 3, 5, and 7) is 23. Theorem for solving simultaneous congruences Sunzi's original formulation: x ≡ 2 (mod 3) ≡ 3 (mod 5) ≡ 2 (mod 7) with the solution x = 23 + 105 k, with k an integer
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