The Chinese Remainder Theorem

The Chinese Remainder Theorem A presentation by Vincent Kusiak

Origins • First developed in the third century A.D. by Chinese mathematician Sun Tzu- unfortunately not the same as his more famous countryman who authored the Art of War • Re-published in a 1247 book by Qin Jiushao, titled “Mathematical Treatise in Nine Sections”

Modular Arithmetic • The theorem only applies in modular arithmetic. • Specifically, when we have the relationship x = k mod j, where x minus k equals some number which is a multiple of j. • For example, 38 = 14 mod 12 • We can see, then that k is broken up into multiple ‘classes’, with any given k +/- j

So what the heck is it?

Well first… • You need a set of numbers that are pairwisecoprime integers; {9,10,19} or {10,7,33,13} • That is, a set of numbers that, when paired, are only divisible by 1; {9,10} or {33,13} • The theorem states that for any set of these integers n1, n2… nk and any given integers a1, a2… ak, there exists an X such that— • X = a1 mod n1, X = a2 mod n2… X = ak mod nk

A problem— • Just by knowing the truth of the theorem, can you figure out what X must be in the system • X = 2 mod 3 • X = 3 mod 4 • X = 1 mod 5

A simple(ish) proof X = 2 mod 5 gives X = 5k + 2 X = 1 mod 11 X = 4 mod 19 X = 5k + 2 = 1 mod 11 so 5k = -1 mod 11 which via equivalence classes is 5k = 10 mod 11, thus k = 5 mod 11

SO… k = 11j +2 and thus X = 5(11j + 2) + 2 and THUS X = 5(11j) + 12 = 4 mod 19 so 5 * 11 * j = -8 mod 19 so 5 * 11 * j = 11 mod 19 So 5j = 1 mod 19 And j = 4 mod 19 because 4 * 5 -1 = 19 So j = 19m + 4 and so X = 5 * 11 * 19m +5 * 11 * 4 +12 Phew.

The Chinese Remainder Theorem