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39 - Chapter 3 Review:9...

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Unformatted text preview: Chapter 3 Review :9 15.1fesndhareintegers,thenntationn|hetsmtsfor .mdtbenotationnfbstsndsfor 16. IT. 13. 19. 22. 2?. 55‘5“ According to the theorem about difiSihilitF by aprime number, given any integer n 3:» 1, there is a . The unique factorization theorem [fundamental theorem of arithmetic} sears that given any htegern}!,neanbewrittenasa inawavthatisunique,exoeptpom'blvfertbe___ inttrlrlilehthetournamentlill‘eill.rril:t~er:|+ The qunttent-rernainder theorem save that given an},r integer n and any positive integer ii, there exists unique integereqandrsuehtbat . Enhannnnegativeintegeranddisapnsitiveinteger,thenndied=_andamedd= where . Cl‘heperit}r property says thatanyintegeriseither . 2L Supposethatatsmnepointhaproofvouknowthetoneofthestetementeitlcud-gorilla istmeandynuwsnttoshowthatregardlessofwhiehstatementhappenstnbetrueaoertain conclusion Gerillfollnw. Tbenyouneedtoshewthat and and . .__, Given surf.r real number at, the floor of at is the unique integer n sueb that _. . Given any real number :s. the ceiling of :r is the unique integer n such that . . 'Ib prove a statement by oontradictlern, you suppose that and you show that _. . To prove a statement of the form We: E B, if Pile} then Q[x}” by rssitrapositionI you suppose that andyousbowthet . . Dneeravtoprovethst fiisanbrationalnumberistoaesumethst Ji=afbfnrsomeintegers nandbwith no oommnnfaotorsgrester than 1, use the lemma that save that ifthe square of anintegeriseventhen ,endeventuallyshnwtbatnandb . Dnewtoprovethatthemsreinflnitelymamrpflmenumbereistoassumethat thereare eel:r finitelvmanypflme numbers p1,pg,...,pmennstruetthenumber_, and thensbnvv thatthisnumberhaetobedivislblebyaprimenumberthatisgreeterthan . Whenanalgorithmfitatementoftheformzzeiseieecuted, . . lEonsilier an algorithm statement of the following form. if [modifien] then 31 else a: When such a statement is executed. the truth or faltfityr of the maditien is- evaluated- If mnfifinn is true, . If mastitien is false. .Gonsidersnalgoritbmstatementoftbefinllowingform. while {Man} fstatsmmflt'mt make up the body of the loop} and. while Wbensuchastatementisexeented,thetnithorfslsltjrofthenendtfinnisevaluated. If mnditinnietrue, .Ifmdtflenisfalss, . ...
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