# Blog

## What is the Chinese Remainder Theorem? ## How do you find the system of congruence using the Chinese remainder?

• The following is a general construction to find a solution to a system of congruences using the Chinese remainder theorem: . y i = N n i = n 1 n 2 ⋯ n i − 1 n i + 1 ⋯ n k. . are pairwise coprime). . v v to the system of congruences. Then u ≡ v ( m o d n 1 n 2 ⋯ n k). ). .

## Is there a proof of Sunsun-Tzu's Chinese Remainder Theorem?

• Sun-tzu's work contains neither a proof nor a full algorithm. What amounts to an algorithm for solving this problem was described by Aryabhata (6th century). Special cases of the Chinese remainder theorem were also known to Brahmagupta (7th century), and appear in Fibonacci 's Liber Abaci (1202).

## What is the remainder when divided by 60?

• In the above example, the product is 3*4*5 = 60. And 11 is one solution, other solutions are 71, 131, .. etc. All these solutions produce the same remainder when divided by 60, i.e., they are of form 11 + m*60 where m >= 0.  