E36 -- Solutio problematis arithmetici de inveniendo numero, qui per datos numeros divisus relinquat data residua
(Solution of problems of arithmetic of finding numbers which, when divided by given numbers, leave given remainders)
Summary:
Euler proves the Chinese Remainder Theorem by constructing an algorithm to find the smallest number which, divided by given numbers, leaves given remainders. He begins by solving the case in which two relatively prime divisors with corresponding remainders are given and proposes that by repeating his algorithm, he can solve similar problems with any number of constraints. Euler then discusses scenarios in which divisors are not relatively prime, and ends the paper with an application of his algorithm to a classic problem: dating events in Roman indictions.
Publication:
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Originally published in Commentarii academiae scientiarum Petropolitanae 7, 1740, pp. 46-66
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Opera Omnia: Series 1, Volume 2, pp. 18 - 32
- Reprinted in Comment. acad. sc. Petrop. 7, ed. nova, Bononiae 1748, pp. 43-60 [E36a]
- Reprinted in Commentat. arithm. 1, 1849, pp. 11-20 [E36b]
- Republished in L. Euler, Cours d’arithmétique raisonnée, Paris 1865, pp. 437-459 [E36Aa]
Documents Available:
- Original publication: E036 (in the Commentarii)
- English translation (Crystal Lai, Ben Russell, and Kathy Yu, all of Carleton College, under the supervision of Stephen Kennedy): E036
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