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The oldest remainder problem in the world was first discovered
in a third

Century
Chinese mathematical treatise entitled Sun Zi Suanjing l*T-lJ~(The

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Mathematical
Classic of Sun Zi), of which the author was unknown.

Nowadays,
the remainder problem in Sun Zi Suanjing is popularly known as

the
Chinese Remainder Theorem, for the reason that it first appeared in a Chinese
mathematical treatise.

 

 Since its
first appearance in Sun Zi Suanjing, the Chinese Remainder

Theorem
continued to attract many ancient mathematicians from other

civilizations
to discuss and give commentary of it in their respective

mathematical
treatises. The purpose of this paper is to discuss the

development
of the interesting Chinese Remainder Theorem in the ancient

civilizations,
since Sun Zi Suanjing.

 

In
modern form, the remainder problem and the solution given in the Sun Zi

Suanjing
can be written as:

Remainder
Problem:

x = 2(mod3)= 3(mod5)= 2(mod7),

where
xis an unknown that satisfies the requirements given in the remainder

problem
and needs to be determined

 

During
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,
his problem

Was  x

 a(mod 3)

 b(mod 5)

 c(mod 7), which
was not entirely the same as

Sun Zi
Suanjing. It was clear that Ibn Tahir had advanced further in his

discussion
of the remainder problem where arbitrary remainders, a, b, and c

were
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.

 

The
popular Chinese Remainder Theorem found its way to Europe in a

famous
mathematical treatise by Italian mathematician Leonardo Fibonacci in

1202
entitled Liber Abaci

 

Fibonacci
was the first European to pave the way for the discussion of

the
Chinese Remainder Theorem in his treatise. Later, in the 14th and 15th

century
Isaac Argyros and Frater Frederius discussed the remainder problem

in
their treatises Eisagog·e Arithm’etik’e and Munich Manuscript respectively.

It was
not surprising that the discussion of the remainder problem in the treatises of
Argyros and Frederius was similar to that of Ibn Tahir.

In
general, many European mathematical treatises in antiquity were greatly

influenced
by the works of the Muslim mathematicians.

The
chronological development of the Chinese Remainder Theorem in the

ancient
mathematical treatises is summarized as follow:

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