1. Describe the history of the Chinese Remainder Theorem. Describe some of the relevant problems posed in Chinese and Hindu writings and how the Chinese Remainder Theorem applies to them. Please show references.
According to D.Wells, the following problem was posed by Sun Tsu Suan-Ching (4th century AD):
There are certain things whose number is unknown. Repeatedly divided by 3, the remainder is 2; by 5 the remainder is 3; and by 7 the remainder is 2. What will be the number?
Oystein Ore mentions another puzzle with a dramatic element from Brahma-Sphuta-Siddhanta (Brahma's Correct System) by Brahmagupta (born 598 AD):
An old woman goes to market and a horse steps on her basket and crashes the eggs. The rider offers to pay for the damages and asks her how many eggs she had brought. She does not remember the exact number, but when she had taken them out two at a time, there was one egg left. The same happened when she picked them out three, four, five, and six at a time, but when she took them seven at a time they came out even. What is the smallest number of eggs she could have had?
Problems of this kind are all examples of what universally became known as the Chinese Remainder Theorem. In mathematical parlance the problems can be stated as finding n, given its remainders of division by several numbers m1, m2, ..., mk:
n = n1 (mod m1)
n = n2 (mod m2)
n = nk (mod mk)
The modern day theorem is best stated with a couple of useful notations. For non-negative integers m1, m2, ..., mk, their greatest common divisor is ...
History, Appearance and Application of Chinese Remainder Theorem in Chinese and Hindu Writings is discussed. The solution is detailed and well presented.