Download 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 Tzuunfortunately 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.