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

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: