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Mike Eden (5)

Midterm Notes

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MATH 135
Mike Eden

Math 135 Kathy Liu 1Propositions and DefinitionsDivisibility An integer m divides an integer n and we write mn if there exists an integer k so that nkm a0 since 00x a This is true for a0 as wellFor all nonzero integers a 0 does not divide a since there is no integer k so that k x 0aProposition Transitivity of Divisibility Let a b c 2 Z If ab and bc then ac Proposition Divisibility of Integer Combinations DIC Let a b and c be integers If ab and ac then abxcy for any x yZProposition Bounds By Divisibility BBD If ab and b 6 0 then abProposition Division Algorithm DA If a and b are integers and b0 then there exist unique integers q and r such that aqbr where 0rbGreatest Common Divisor Let a and b be integers not both zero An integer d0 is the greatest common divisor of a and b written gcdab if and only if1 da and db common and 2 if ca and cb then cd greatest For a0 the defn implies gcda 0a and gcda aa We define gcd0 0 as 0Proposition GCD With Remainders GCD WR If a and b are integers not both zero and q and r are integers such that aqbrthen gcda bgcdb r Proposition Divisibility of Integer Combinations Let a b and c be integers If ab and ac then abxcy for any x yZEuclidean Algorithm EA Basically a chain of GCDwR where you used gcdab gcdb r to break a large number down The last nonzero number is the GCD see slides for further clarificationProposition GCD Characterization Theorem GCD CT If d is a positive common divisor of the integers a and b and there exist integers x and y so that axbyd then dgcda b
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