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Mike Eden (5)

full list of algebra propositions

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Department
Mathematics
Course
MATH 135
Professor
Mike Eden
Semester
Fall

Description
1Propositions1Transitivity of Divisibility TDLet a b c If ab and bc then acProofSince ab by definition of divides there exists an integer k such that akb Since bc by definition of divides there exists an integer h such that bhcSubstitute ak into b and akhcSince kh is an integer it follows that ac2Divisibility of Integer Combinations DICLet a b c If ab and ac then abxcy for any x yProofSince ab there exists an integer k such that ak bSince ac there exists an integer h such that ahcLet x and y be any integersSubstitute b and c for ak and ahbxcyakxahyakxhysince kxhy is an integer it follows that abxcy since akxhybxcy23Bounds By Divisibility BBDIf ab and b0 thenabProofSince ab there exists an integer k such that akbSince b0 k0 thenk 1 bkaaThereforeba 4Division Algorithm DAIf a b and b0 then there exists unique integers q and r such that aqbr where 0rb202 x 76213 x 70203 x 715GCD With Remainders GCD WRIf a b where a b0 and q r are integers such that aqbr then gcd a bgcd b rProofLet dgcd a bSince dgcd a b by definition of gcd dbSince raqb by the Division Algorithm da and db daqb by DICHence dr and d is a common divisor of b and rLet c be a divisor of b and r Since cb and cr cqbr by DICSince aqbr by Division Algorithm caBecause dgcd a b ca and cb it follows that cdTherefore dgcda bgcdb r
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