Abstract
Sun-Tsu wrote the treatise Sunzi Suanjiing around the 3rd century. The problem of finding an integer x which is simultaneously 2 modulo 3, 3 modulo 5 and 2 modulo 7 was considered. The smallest solution was found to be 23 and such a result is now called the Chinese Remainder Theorem (CRT). From early times–perhaps, from the 1st century itself–the CRT was employed in the preparation of calendars. In India, Aryabhata’s mathematics from the 5th century contains instances of the CRT. However, a multivariable version of CRT does not seem to be well known and is not a part of textbooks. Qin Jiushao seems to have considered one such version in the 13th century. In this article, the basic CRT is recalled and some multivariable versions are studied using elementary linear algebra.
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Oystein Ore, The general Chinese remainder theorem, The American Mathematical Monthly, Vol.59, No.6, pp.365–370, 1952.
Fredric T Howard, A generalized Chinese remainder theorem, The College Mathematics Journal, Vol.33, No.4, pp.279–282, September, 2002.
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B Sury was in the School of Mathematics of TIFR Bombay from 1981 to 1999. Since 1999, he has been with the Indian Statistical Institute in Bangalore. His research interests are in algebra and number theory. He is the Karnataka coordinator for the Mathematical Olympiad Programme in India.
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Sury, B. Multivariable Chinese remainder theorem. Reson 20, 206–216 (2015). https://doi.org/10.1007/s12045-015-0171-x
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DOI: https://doi.org/10.1007/s12045-015-0171-x