Naive Grouping & Trial Multiplication for Factorization (Notes)

8/10/2019 Naive Grouping & Trial Multiplication for Factorization (Notes)

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[YL@OCOAEY_ AB^LXNB AB^ NOX^YOI\^L

Aevl Mrbupoam & ^roe` Cu`top`ohetoba dbr Dehtbrozetoba

Iy,

Drln Tob`l

BTTB Doaeahoe` Xystlcsdrln.vob`lKbvvbdoaeahoe`systlcs.hbc

Od et dorst tgl onle os abt eisurn,tgla tglrl os ab gbpl dbr ot.

E`ilrt Loastloa
mailto:[email protected]:[email protected]:[email protected]


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5 | [ e m l

O gevl illa tgoajoam e `bt eibut dehtbroam ètl`y. O doan tgl prbi`lc tb il novosoba. Aecl`y ybu

gevl tb esj gbw ceay mrbups bd mb oatb . ^gos nbls abtgoam dbr easwlroam tgl qulstoba6]get os tgl dbr eay movla 2 ^glrl ges tb il e wey tget wo`` tl`` ybu gbw ceay mrbups bdmbls oatb ot. O tgoaj O dbuan ot, tgus evbonoam troe` novosoba dbr dehtbrozetoba.

O wes pèyoam wotg e ceri`l slt wotg tgl jons ean ot got cl. _bu hea suhhlssovl`y mrbup uato` otmovls ybu ea easwlr bd gbw ceay mrbups bd mb oatb . Dbr lxecp`l, wgla dehtbroam >?6

Xpòt >? oatb 9 mrbups, es wbu`n il eay plrsba's oaotoe` spòt (O tgbumgt bd tgl

eaholats ean rbhjs wotg tgos) `levoam ybu wotg 9< ean 9>.

Xpòt 9< oatb 58 ean 59

Xpòt 9> oatb 59 ean 59

Xpòt 58 oatb 4 ean =

Xpòt 59 oatb 4 ean 4

Xpòt 59 oatb 4 ean 4 Xpòt 59 oatb 4 ean 4

^glrl erl = mrbups bd 4 ean 5 mrbup bd =. [èhl 5 drbc tgl mrbup bd = oatb tgl

mrbups bd 4, ean ybu abw gevl = mrbups bd = dbr tgl dehtbrs bd >?. Et lehg spòt,tglrl os e novosobastlp tb il nbal, splhodohe``y oa tgos lxecp`l, hea tgl mrbup bd58 il spòt oatb tgl rlceoaoam 8 mrbups edtlr tgl dorst spòt2 br hea tgl mrbup bd =

il lvla`y pèhln oatb tgl rlceoaoam = mrbups bd 42

^gl stlps bd tgl prbhlnurl alln tb il dbrceòzln durtglr, iut od tgos hea il tlstln ba e sce``lr

YXE aucilr ot's pbssoi`y e mecl hgeamlr. Ean ab, ot's abt e dehtbr trll, wgohg occlnoetl`y

novonls edtlr lehg Abnl, ot os cbrl ejoa tb e mrbupoam trll.

Aucilrs novosoi`l iy 8 et tgl slhban abnl drbc e slln mrbup bd 96

48

85 89

5< 54 54 54

5< os novosoi`l iy tgrll ean < erl pèhln oa lehg mrbup, `levoam tgl dehtbrs6 8 mrbups bd 95.

= 4< 4< 4 Abnl < Abnl 4 Abnl = Abnl 7 Abnl ?

Abnl

5;

9 > 7 54 89 4> 597 9 >7 ?4 5?9 87> =47 5; 7; 54; 89; 4>; 597; 9 97 >>7 7?4 5=?9 8? 57 84 =9 5>> 977 4;7

Bivobus`y tglrl os ab alln tb stbrl tgl mrbupoams, retglr fust onlatody tglor hbuat ean veùl (es

nlcbastretln oa tgl ocemls il`bw). ^gl db``bwoam ocemls rlprlslat du`` slqulahls oa Lxhl`.

^gly e`` gevl e splhodoh ru`l oavb`voam tgl d`bbr ean tgl hloòam bd tgl prlvobus hbuat ean veùls.

Emeoa, tglrl os ab alln tb stbrl tgl du`` slqulahl, fust tgl èst hbuat ean veùl sb tget tgl alxt

hea il mlalretln.


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> | [ e m l

Lxecp`l bd tgl sbùtoba movla e splhodoh slln mrbup, ean gbw tgl rlst bd tgl slqulahl d`bws drbc tget oaotoe` sbùtoba.

Oa tgos lxecp`l, movla e slln mrbup tbte` bd 5>?, gbw nb ybu spòt 3 77,9=5usoam hbuat veùls


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< | [ e m l

Ean wl hea sll ba tgl 8 drbc 5>7, wl gevl e cbn bd ; dbr 5;8

mrbups ean 457 clcilrs bd tgl btglr mrbup bd >;. ^gus 5;8 os e dehtbr bd 77,9=5.


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4 | [ e m l

Ean wotg 54 novosobas (oahrlesoam ean nlhrlesoam drbc lvlry ;.;49


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= | [ e m l

Dbr 3 5;,;>5,;? ot ba`y tbbj 5,55? oahrlclats tb doan tgl dehtbr ?479/9 3 5 wbu`n gevl e`sb prbnuhln tgl dehtbr ?


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7 | [ e m l

Es hbaflhturln, dbr aucilrs wotg dehtbrs h`bsl tb bal eabtglr, tgl otlretobas erl `bw. Oa tgos hesl, 3 57


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? | [ e m l

@lcce(5)6 E`` slln mrbups tb wo`` hbvlr e`` bd tgl sllns. Durtglrcbrl, ba`y tgl bnn sllnmrbups erl alhlssery mrletlr tgea 9.

Dbr lxecp`l, oa dehtbroam

3


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5; | [ e m l

Movla tgl oasomaodoheat hbcputetobas tb rua tgl slqulahl drbc e movla slln, tgl lddoholahy cey

rlst oa pertotobaoam ba`y tgl bnn oatlmlrs wotgoa Z9,S. Dbr lxecp`l, usoam ceyyol`n ea epprbproetl aucilr bd pertotobas

\soam ea lxecp`l h`bsl tb 5;,;;; tb o`ùstretl, wgla dehtbroam

3 5;,>;8tgl db``bwoam mrbups

wbu`n alln tb il slln mrbups oa brnlr tb lasurl e`` erl hglhjln6

3 5 >8 >< >= >? ;8.

@lamtg bd slqulahls6

]l clatobaln leròlr tget tgl slqulahl sgbu`n stbp bahl tgl mrbup aucilrs erl mrletlr tgea br

lque` tb tgl hbuats wotgoa tgl mrbups. ^gos lddlhtovl`y huts tgl slqulahl oa ge`d ean ennrlssls

sbcl hbahlras dbr dehtbroam vlry èrml aucilrs.

Xoahl tgl mrbup aucilr nbui`ls lvlry Abnl, tgl aucilr bd Abnls os rlètln tb tgl iot sozl bd tgl

aucilr. Dbr lxecp`l, 5;; Abnls erl alhlssery tb rlehg 9, 3 5.94=4


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55 | [ e m l

Ea lxecp`l wo`` gl`p o`ùstretl tgos ocpbrteat dleturl bd òcotln slqulahl `lamtg6

OAHYLEXOAM XB@\ÔBA DYBC ;

XLLN Cbn 5

OAHYLEXOAM XB@\ÔBA DYBC ;

XLLN Cbn 9,;7;.4; A\CILY 5;,>;8

< 5 9 9;7;

5 8 9;75

5; < = 5;>;

9 8 5;>5

9; 55 5=


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59 | [ e m l

[rbieio`oty bd e Dehtbr

\soam 3 =?7,4;=, wgohg os (5;5)(=?;=), 3 7?8.4>7 wl nlcbastretl tgl cu`totunl bdslln mrbups tget wo`` yol`n e dehtbr.

OAHYLEXOAM XB@\^OBA DYBC ;XLLN Cbn 5,955.7< A\CILY =?7,4;=

4


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58 | [ e m l

OD dehtbrs erl h`bsl tgla tgos cltgbn wo`` doan tglc oa e rlesbaei`l ecbuat bd tocl. Od

dehtbrs erl der tgla ot wo`` tejl `bamlr, iut et tgl ilaldot bd e rlnuhln cexocuc pbssoi`l

otlretoba. OD dehtbrs erl vlry der epert, tgly ilhbcl quohj tb doan emeoa. Xb lxtrlcl`y

h`bsl ean lxtrlcl`y der epert dehtbrs erl lesy. Movla YXEs prbplasoty tb mlalretl tglèrmlst aucilr, ot os hbaflhturln tget tgl dehtbrs wo`` il rlètovl`y h`bsl tb bal eabtglr,

`lenoam tb e rlesbaei`l ecbuat bd tocl tb dehtbr usoam tgos cltgbn.

3 7

Nlabcoaetbr lxpèoaln6

x9 tb rlprlslat tgl

upplr ibuan.

x9 tb rlprlslat oahrlesoam ean nlhrlesoam drbc tgl pertotobas.

x9 tb rlprlslat fust tgl bnn aucilrs.

YXE Lxecp`ls

YXE-5;; 3 5


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5> | [ e m l

YXE-55; 3 898>5=?=9??57;=7899979945?

87>4=;94==>55?8=599


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5< | [ e m l

\ajabwa @bwlr Xlln Mrbups

^gl eibvl YXE lxecp`ls wlrl o`ùstretoam jabwa slln mrbups. Gbwlvlr, tglrl lxost uajabwa

`bwlr slln mrbups, durtglr oahrlesoam tgl prbieioòty bd doanoam e dehtbr.

3 4?,??,945(coaus 5) ean ots `bwlr dehtbr slln mrbup bd =?;=.

NLHYLEXOAM XB@\ÔBA DYBC .9,4>7.=7

9>,779.?8

XLLN Cbn

=,8== 58 779

= ?>778

5>= ; =?;= >=>>5

48


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54 | [ e m l

OAHYLEXOAM XB@\^OBA .957=,=8? 55;4??

5>;8 5, 55;=;;

594>4 8995 >=8?


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5= | [ e m l

E`tlraetovl ^b Novosoba E`tbmltglr

Bal e`tlraetovl os tb novosoba oa eay dbrc os tb usl cu`topòhetoba. ^gl oasomgts meralrln drbc tgl

oaotoe` sbùtoba tb tgl slln mrbup e``btclat yol`nln e troe` cu`topòhetoba cltgbn.

Xtertoam drbc tgl d`bbr ean tgl hloòam bd tgl squerl rbbt bd tgl aucilr, wl hea `bwlr tgl d`bbr

ean reosl tgl hloòam suhg tget wgla cu`topòln wl hea hbcperl tb. \soam e sce`` lxecp`l wo``hbadorc tgl oatuotoba

3 5,79?

]glrliy,

3 >9 3 >8

Xtlp 56

>5>5 * >8 3 5,=48

1 5,=48sb wl enn tb tgl hloòam

Xtlp 96

>=>5 * >= 3 5,?9=

0 5,?9=sb wl nlhrlesl tgl d`bbr

Xtlp 86

8?8? * >= 3 5,788

0 5,788sb wl nlhrlesl tgl d`bbr, emeoa

Xtlp >6

8=8= * >= 3 5,=8?

1 5,=8?sb wl enn tb tgl hloòam

Xtlp ?

8= * >? 3 5,758 1 5,758sb wl enn tb tgl hloòam emeoa

Xtlp 46


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57 | [ e m l

Xtlp =6

8888 *


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5? | [ e m l

]l erl dbrhln tb utoòzl lehg hbcpbalat slqulahl bd tgl aba-lque` peors usln es tgl d`bbr, soahl

ab nlhosoba ru`l ylt lxosts tb enn oasomgt tb wgohg slqulahl os tgl d`bbr slqulahl.

Xb oa bur probr lxecp`l 3 5,79?, stertoam wotg >5, bur dorst peoroam wbu`n gevl illa >5*>?wgohg lque`s 9,;;?. Bur slhban ettlcpt oa tgl 5 slqulahl wbu`n il 85*>? wgohg os tbb `bw,

tgla ennoam tb tgl hloòam oa tgl ? slqulahl, wl gevl 85*,49? 995,45;

OD [U 1 A, YLN\HL [ \AÔ@ [U 0 A OD [U 0 A, OAHYLEXL U \AÔ@ [U 1 A

[ U [U A-([U) [ U [U A-([U

5=,94= 5=,85= 9??,;59,48? (9?4,>

5=,94? (598,98;)

5=,98= 5=,85= 9?7,>?8,59? 998,55;

5=,98= 5=,89= 9?7,44??

5=,98= 5=,8>= 9??,;5;,98? (9?>,;

5=,9;= 5=,8>= 9?7,>7?,79? 994,>5;

5=,9;= 5=,8

5=,9;= 5=,8== 9??,;;4,;8? (97?,7


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9; | [ e m l

[ U [U A-([U) [ U [U A-([U

5=,5?= 5=,8== 9?7,789,94? (554,;8;)

5=,5== 5=,8== 9?7,>7>,=9? 985,


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95 | [ e m l

[ U [U A-([U) [ U [U A-([U

54,=7= 5=,=4= 9?7,9,49? >45,45;

54,=7= 5=,=== 9?7,>99,>?? 9?8,=

54,=7= 5=,=?= 9?7,=9,;;

54,=== 5=,=?= 9?7,;

54,== 5=,79= 9?7,7,=4? 54=,>=;

54,=>= 5=,78= 9?7,=54,98? -

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Naive Grouping & Trial Multiplication for Factorization (Notes)