Multivariable Chinese remainder theorem

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.