Introductions to ExtendedGCD

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Edmund Aldous Turner
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1 Itroductios to ExtededGCD Itroductio to the GCD ad LCM (greatest coo divisor ad least coo ultiple) Geeral The legedary Greek atheaticia Euclid (ca BC) suggested a algorith for fidig the greatest coo divisor of two itegers, which was later aed the Euclidea algorith. This algorith, as well as coputatioally refied versios, are i widespread use today. Defiitios of GCD ad LCM The GCD ad LCM group of fuctios icludes the followig three fuctios: the greatest coo divisor (gcd): gcd 1, 2,, the least coo ultiple (lc): lc 1, 2,, the exteded greatest coo divisor (egcd): egcd 1, 2,, These fuctios are defied i the followig ways: gcd 1, 2,, p ; p k p 1 k q q q p k q 1 k gcd 1, 2,, p ; Re p I p Re k 1 k q q p Re q I q Re k p I k q p I k q 1 k lc 1, 2,, p ; p p k 1 k q q p p q k 1 k lc 1, 2,, p ; Re p I p Re p k I p k 1 k q RAbs q p Re q I q Re q k I q k 1 k egcd 1, 2,, gcd 1, 2,,, r 1, r 2,, r ; gcd 1, 2,, 1 r 1 2 r 2 r Re k I k Re r k I r k 1 k. gcd 1, 2,, is the greatest coo divisor of the itegers (or ratioal) k. It is the greatest iteger factor coo to all the k, 1 k. lc 1, 2,, is the least coo ultiple of the itegers (or ratioal) k. It is the iial positive iteger that divides all the k, 1 k.

2 2 egcd 1, 2,, is the exteded greatest coo divisor of the itegers k. I particular, egcd, gcd,, r, s ; gcd, r s Re I Re I Re r I r Re s I s. Exaples: The greatest coo divisor of 21 ad 48, gcd 21, 48 is 3. Siilar exaples are gcd 27, 48, 36 3, gcd 27 3, , gcd 2 3, The least coo ultiple of the three ubers 2, 4, ad 5, lc 2, 4, 5 is 20. Siilar exaples are lc 27, 48, , lc 27 3, , lc 2 3, The exteded greatest coo divisor of 21 ad 48 egcd21, 48 is 3, 7, 3 because the greatest coo divisor gcd 21, 48 3 ad Siilarly, egcd 15 9, 5 7 1, 2 4, 7 6 ; 1 gcd 15 9, Coectios withi the group of the GCD ad LCM ad with other fuctio groups Represetatios through equivalet fuctios The fuctios gcd 1, 2,, ad lc 1, 2,, satisfy the followig iterrelatios: gcd, lc, ; gcd 1, 2,, k1 1 k2 k 1 1 k1 1 lc, gcd, ; k1 k1 1 lc k1, k2 k1 1 lc,, k gcd, k, k k ;,, k k2 k 1 1 k2 k 1 1 k3 k 2 1 lc k1, k2, k3 k3 k 2 1 gcd lc k,, lc k,, lc, lc gcd k,, gcd k,, gcd, ;,, k. k4 k 3 1 lc k1, k2, k3, k4 The best-kow properties ad forulas of the GCD ad LCM Specific values for specialized variables The fuctios GCD ad LCM gcd 1, 2,,, egcd 1, 2,,, ad lc 1, 2,, have the followig values for specialized values:

3 3 gcd egcd lc gcd egcd, sg lc gcd0, egcd 0,, 0, sg lc 0 0 gcd, egcd,, 0, sg lc, gcd, egcd,, 0, sg lc, gcd 1, 1,, 1 1 gcdp 1, p 2 1 ; p 1 p 2 p 1 p 2 gcd, lc, ; gcd, lc p, q lc gcd, p, gcd, q ; p q gcd lc, p, lc, q lc, gcd p, q ; p q gcd lc k,, lc k,, lc, lc gcd k,, gcd k,, gcd, ; k gcd od, gcd, ; gcd 2 1, gcd, 1 ; gcd F, F F gcd, ;. egcd 1, 2,, p 1, 1, 2,, p 1, sg 1 ; 1 2 p 1 2 p 1 0 lc 1, 1,, 1 1 lcp 1, p 2 p 1 p 2 ; p 1 p 2 p 1 p 2 lc, gcd, ; lc, gcd p, q gcd p q lc gcd, p, gcd, q p q lcgcd k,, gcd k, gcd lc k,, lc k k The first values of the greatest coo divisor (gcd(, )) of the itegers ad for 1 20 ad 1 20 are described i the followig table:

4 4 \ The first values of the exteded greatest coo divisor (egcd, ) of the itegers ad for 1 10 ad 1 10 are described i the followig table: \ , 0, 1 1, 1, 0 1, 1, 0 1, 1, 0 1, 1, 0 1, 1, 0 1, 1, 0 1, 1, 0 1, 1, 0 2 1, 0, 1 2, 0, 1 1, 1, 1 2, 1, 0 1, 2, 1 2, 1, 0 1, 3, 1 2, 1, 0 1, 4, 1 3 1, 0, 1 1, 1, 1 3, 0, 1 1, 1, 1 1, 2, 1 3, 1, 0 1, 2, 1 1, 3, 1 3, 1, 0 4 1, 0, 1 2, 0, 1 1, 1, 1 4, 0, 1 1, 1, 1 2, 1, 1 1, 2, 1 4, 1, 0 1, 2, 1 5 1, 0, 1 1, 1, 2 1, 1, 2 1, 1, 1 5, 0, 1 1, 1, 1 1, 3, 2 1, 3, 2 1, 2, 1 6 1, 0, 1 2, 0, 1 3, 0, 1 2, 1, 1 1, 1, 1 6, 0, 1 1, 1, 1 2, 1, 1 3, 1, 1 7 1, 0, 1 1, 1, 3 1, 1, 2 1, 1, 2 1, 2, 3 1, 1, 1 7, 0, 1 1, 1, 1 1, 4, 3 8 1, 0, 1 2, 0, 1 1, 1, 3 4, 0, 1 1, 2, 3 2, 1, 1 1, 1, 1 8, 0, 1 1, 1, 1 9 1, 0, 1 1, 1, 4 3, 0, 1 1, 1, 2 1, 1, 2 3, 1, 1 1, 3, 4 1, 1, 1 9, 0, , 0, 1 2, 0, 1 1, 1, 3 2, 1, 2 5, 0, 1 2, 1, 2 1, 2, 3 2, 1, 1 1, 1, 1 The first values of the least coo ultiple (lc, ) of the itegers ad for 1 20 ad 1 20 are described i the followig table:

5 5 \ Aalyticity The fuctios gcd 1, 2,, ad lc 1, 2,, are oaalytical fuctios defied over with values i. The fuctio egcd 1, 2,, is a vector-valued oaalytical fuctio defied over. Periodicity All three fuctios gcd 1, 2,,, egcd 1, 2,,, ad lc 1, 2,, do ot have periodicity. Parity ad syetry The fuctios gcd 1, 2,, ad lc 1, 2,, are eve fuctios: gcd 1, 2,, gcd 1, 2,, gcd 1, 2,, gcd 1, 2,, lc 1, 2,, lc 1, 2,, lc 1, 2,, lc 1, 2,,. The fuctios gcd 1, 2,, ad lc 1, 2,, have perutatio syetry: gcd, gcd, gcd 1, 2,, k,, j,, gcd 1, 2,, j,, k,, ; k j k j lc, lc,

6 6 lc 1, 2,, k,, j,, lc 1, 2,, j,, k,, ; k j k j. Series represetatios The fuctio gcd, has the followig su represetatios: 1 gcd, 2 k 1 k 2 gcd, k 1 k 2 2 k 1 k, where p is the floor fuctio ad q is the Kroecker delta fuctio. Product represetatios The fuctios gcd 1, 2 ad lc 1, 2 have the followig product represetatios: j k gcd 1, 2 j 1 i Α p 1, j,α 2, j i,j ; 1 2 factors k p k,1, Α k,1,, p k,jk, Α k,jk p k,j Α k,j 1 k 2 j k lc 1, 2 j 1 ax Α p 1, j,α 2, j i,j ; 1 2 factors k p k,1, Α k,1,, p k,jk, Α k,jk p k,j Α k,j 1 k 2. Geeratig fuctios The fuctio gcd k, ca be represeted as the coefficiets of the series expasio of correspodig geeratig fuctios, which icludes a su of the Euler totiet fuctio: gcd k, x k Φ d. 1 x d k 1 d Trasforatios with ultiple arguets x d The GCD ad LCM fuctios gcd 1, 2,,, egcd 1, 2,,, ad lc 1, 2,, satisfy special relatios icludig ultiple arguets, for exaple: gcd p 1, p 2,, p p gcd 1, 2,, ; p gcd Μ, Ν gcd, gcd Μ, Νgcd Μ Ν. gcd,, Ν gcd Μ, Ν gcd gcd,, Μ gcd Μ, Ν ; Idetities

7 7 The GCD ad LCM fuctios satisfy soe parallel idetities that ca be preseted i the fors show i the followig table: gcd gcdgcd,, p gcd, gcd, p gcd 1, gcd 2, 3,, gcd 1, 2, 3,, gcd,, p gcd, gcd, p lc lclc,, p lc, lc, p lc 1, lc 2, 3,, lc 1, 2, 3,, lc,, p lc, lc, p Suatio There are ay fiite ad ifiite sus cotaiig GCD ad LCM fuctios, for exaple: F gcd k 1, k 2,, k f d k 1 1 k 2 1 k 1 k k k 1 1,gcd k, 3 2 k 4 1 k 1 1,gcd k, 1,gcd k, 3 1 k 1 k k 2 8 1,gcd k, 7 1 k 1 k k ,gcd b,d 1 b 1 d 1 b d b d 2 3. Liit operatio k 1 d ; F f d d The followig two related liits iclude the fuctio gcd 1, 2,,. The third liit icludes lc( 1, 2,, ): li li 1 2 k 1 1,gcd k,l 6 Π 2 l 1 1 gcd k r 1, k 2,, k r k Ζ r k k 1 1 k 2 1 k r Ζ r 1 Iequalities The fuctios gcd 1, 2,, ad lc 1, 2,, satisfy various iequalities, for exaple:

8 8 k gcd 1, 2,, k lc 1, 2,, k k 1 j gcd 1, 2,, k k 1 lc 1, 2,, k ; j k k lc 1, 2,, 2 2 ;. Applicatios of the GCD ad LCM The GCD ad LCM fuctios have uerous applicatios throughout atheatics, uber theory, sybolic algoriths, ad liear Diophatie equatios.

9 9 Copyright This docuet was dowloaded fro fuctios.wolfra.co, a coprehesive olie copediu of forulas ivolvig the special fuctios of atheatics. For a key to the otatios used here, see Please cite this docuet by referrig to the fuctios.wolfra.co page fro which it was dowloaded, for exaple: To refer to a particular forula, cite fuctios.wolfra.co followed by the citatio uber. e.g.: This docuet is curretly i a preliiary for. If you have coets or suggestios, please eail coets@fuctios.wolfra.co , Wolfra Research, Ic.

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