The oldest remainder problem in the world was first discovered
in a third
Chinese mathematical treatise entitled Sun Zi Suanjing l*T-lJ~(The
Classic of Sun Zi), of which the author was unknown.
the remainder problem in Sun Zi Suanjing is popularly known as
Chinese Remainder Theorem, for the reason that it first appeared in a Chinese
first appearance in Sun Zi Suanjing, the Chinese Remainder
continued to attract many ancient mathematicians from other
to discuss and give commentary of it in their respective
treatises. The purpose of this paper is to discuss the
of the interesting Chinese Remainder Theorem in the ancient
since Sun Zi Suanjing.
modern form, the remainder problem and the solution given in the Sun Zi
can be written as:
x = 2(mod3)= 3(mod5)= 2(mod7),
xis an unknown that satisfies the requirements given in the remainder
and needs to be determined
the 11th century, Muslim mathematician Ibn Tahir al-Baghdadi discussed the
Chinese Remainder Theorem in his treatise AITakmila fi ‘lim ai-Hisab.
The moduli that Ibn Tahir gave were the same as Sun Zi Suanjing. However,
c(mod 7), which
was not entirely the same as
Suanjing. It was clear that Ibn Tahir had advanced further in his
of the remainder problem where arbitrary remainders, a, b, and c
given in his problem . It is interesting to note that Ibn Tahir was the first
mathematician in antiquity to give an explanation regarding why the numbers 70,
21 and 15 were related to the moduli 3, 5 and 7 respectively.
popular Chinese Remainder Theorem found its way to Europe in a
mathematical treatise by Italian mathematician Leonardo Fibonacci in
entitled Liber Abaci
was the first European to pave the way for the discussion of
Chinese Remainder Theorem in his treatise. Later, in the 14th and 15th
Isaac Argyros and Frater Frederius discussed the remainder problem
their treatises Eisagog·e Arithm’etik’e and Munich Manuscript respectively.
not surprising that the discussion of the remainder problem in the treatises of
Argyros and Frederius was similar to that of Ibn Tahir.
general, many European mathematical treatises in antiquity were greatly
by the works of the Muslim mathematicians.
chronological development of the Chinese Remainder Theorem in the
mathematical treatises is summarized as follow: