ChineseRemainderTheorem
YanHuang
• Thereisapileof𝑛 apples.Ifdividethepileintogroupsof3,thereare
2applesleft.Ifdividedintogroupsof7,there4applesleft.Whatis
theminimalvalueof𝑛?
• Thereisapileof𝑛 apples.Ifdividethepileintogroupsof4,thereare
2applesleft.Ifdividedintogroupsof5,there1applesleft.Whatis
theminimalvalueof𝑛?
ChineseRemainderTheorem
Assume𝑛" and𝑛# arecoprime.Let𝑥 bethesolutiontothefollowing
systemsofmoduloidentities
𝑥 = 𝑎" mod 𝑛"
𝑥 = 𝑎# mod 𝑛# .
Then𝑥 = 𝑋# 𝑛# 𝑎" + 𝑋" 𝑛" 𝑎# mod 𝑁,where𝑁 = 𝑛" ×𝑛# and𝑋" 𝑛" +
𝑋# 𝑛# = 1.
MoreGenerally
• ChineseRemainderTheoremestablishesabijectionbetweenℤ1 ×ℤ2
andℤ12 .
Example
Isomorphism
Let𝔾, ℍ begroupswithrespecttotheoperations⋆𝔾 and⋆ℍ .A
function𝑓: 𝔾 → ℍ isanisomorphismif
1. 𝑓 isabijection,and
2. Forall𝑔" , 𝑔# ∈ 𝔾,𝑓 𝑔" ⋆𝔾 𝑔# = 𝑓 𝑔" ⋆ℍ 𝑓(𝑔# ).
Ifthereexistsanisomorphismbetween𝔾 andℍ,wesay𝔾 andℍ are
isomorphicandwrite𝔾 ≃ ℍ.
• ℤ12 isagroupwithrespecttoeitheradditionormultiplication.
• ℤ1 ×ℤ2 isalsoagroup(withrespecttoentry-wisemoduloeither
additionormultiplication).
• ℤ12 ≃ ℤ1 ×ℤ2 .
• moduloadditionisanisomorphismbetweenℤ!" and ℤ! ×ℤ"
• modulomultiplicationisalsoanisomorphismbetweenℤ!" and ℤ! ×ℤ"
ModuloAdditionisanIsomorphismbetweenℤ12 and ℤ1 ×ℤ2
ModuloMultiplicationisanIsomorphismbetweenℤ12 and ℤ1 ×ℤ2
UsingCRTtoSimplifyModuloComputations
• Calculate
2838mod35
UsingCRTtoSimplifyModuloComputations
• Calculate
2838*12345mod35