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Maximvs commvnis divisor et minimvm commvne mvltiplvm

Svmmarivm

Haec lectio maximvm commvnem divisorem et minimvm commvne mvltiplvm dvorvm nvmerorvm natvralivm definit, et docet qvomodo dvobvs modis compvtentvr: per resolvtionem in factores primos et per algorithmvm Evclidis, qvi factores qvaerere non postvlat. Ex algorithmo identitas Bezovtiana dedvcitvr, secvndvm qvam maximvs commvnis divisor combinatio integra dvorvm nvmerorvm est, et ex ea relatio inter vtramqve notionem, natvra nvmerorvm inter se primorvm, simplificatio et additio fractionvm, et plena solvtio aeqvationvm Diophantearvm linearivm, cvm earvm vsv in qvaestionibvs de nvmmis et de mensvris.

Proposita discendi

  1. Maximvm commvnem divisorem et minimvm commvne mvltiplvm per resolvtionem in factores primos compvtare, et relationem inter vtrvmqve comprobare.
  2. Algorithmvm Evclidis eivsqve formam extensam adhibere vt maximvs commvnis divisor et coefficientes identitatis Bezovtianae obtineantvr.
  3. Nvmeros inter se primos agnoscere, et maximo commvni divisore et minimo commvni mvltiplo vti ad fractiones simplificandas et addendas.
  4. Decernere vtrvm aeqvatio Diophantea linearis solvtionem habeat, omnes eivs solvtiones describere, eamqve ad qvaestiones cvm condicionibvs natvralibvs adhibere.

Maximvs commvnis divisor et minimvm commvne mvltiplvm

In lectione de divisibilitate et divisione cvm residvo relatio d∣ad \mid a inter dvos nvmeros natvrales tractata est. Cvm dvo nvmeri simvl considerantvr, ad rem pertinent divisores qvos commvnes habent et mvltipla qvae participant. Omne par nvmerorvm natvralivm aa et bb saltem vnvm divisorem commvnem habet, scilicet 11, et ad svmmvm nvmervm finitvm eorvm, qvia nvllvs divisor nvmeri aa maior est qvam aa; habet praeterea infinita mvltipla commvnia, inter qvae prodvctvm abab. Qvod si ita est, definitio seqvens sensvm habet.

Definitio (maximvs commvnis divisor et minimvm commvne mvltiplvm). Maximvs commvnis divisor dvorvm nvmerorvm natvralivm aa et bb, qvi scribitvr mcd⁡(a,b)\operatorname{mcd}(a, b), est maximvs nvmervs natvralis qvi vtrvmqve dividit. Minimvm commvne mvltiplvm, qvod scribitvr mcm⁡(a,b)\operatorname{mcm}(a, b), est minimvs nvmervs natvralis qvi vtrivsqve mvltiplvm est.

Altervm est minimvm elementvm copiae non vacvae nvmerorvm natvralivm, qvod per principivm bonae ordinationis exsistit. Prior est maximvm elementvm copiae divisorvm commvnivm, qvae non vacva est et a aa circvmscripta; copiam non vacvam nvmerorvm natvralivm svpra circvmscriptam maximvm habere etiam ex bona ordinatione seqvitvr: copia marginvm eivs svperiorvm non vacva est et minimvm MM habet, et MM ad copiam pertinet, qvia alioqvin omne elementvm minvs esset qvam MM et, cvm nvllvs nvmervs natvralis sit inter M−1M - 1 et MM, non maivs qvam M−1M - 1, qvi margo svperior minor qvam MM esset (si M=1M = 1, copia est {1}\{1\} et MM continet). Exempli gratia, divisores commvnes nvmerorvm 1212 et 1818 svnt 11, 22, 33 et 66, et mvltipla eorvm commvnia svnt 3636, 7272, 108108, et sic deinceps:

mcd⁡(12,18)=6,mcm⁡(12,18)=36\operatorname{mcd}(12, 18) = 6, \qquad \operatorname{mcm}(12, 18) = 36

Ex definitione statim casvs qvidam particvlares seqvvntvr: mcd⁡(a,1)=1\operatorname{mcd}(a, 1) = 1 et mcm⁡(a,1)=a\operatorname{mcm}(a, 1) = a; mcd⁡(a,a)=mcm⁡(a,a)=a\operatorname{mcd}(a, a) = \operatorname{mcm}(a, a) = a; et, si aa dividit bb, tvm mcd⁡(a,b)=a\operatorname{mcd}(a, b) = a et mcm⁡(a,b)=b\operatorname{mcm}(a, b) = b. Ambae operationes praeterea symmetricae svnt: ordo nvmerorvm eventvm non mvtat.

Compvtatio per resolvtionem in factores primos

Per theorema fvndamentale arithmeticae omnis nvmervs natvralis maior qvam 11 vno tantvm modo in factores primos resolvitvr. Expedit aa et bb eisdem primis p1,…,pkp_1, \ldots, p_k scribere, exponentibvs nvllis admissis pro primis qvi in altero tantvm apparent:

a=p1α1p2α2⋯pkαk,b=p1β1p2β2⋯pkβka = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k}, \qquad b = p_1^{\beta_1} p_2^{\beta_2} \cdots p_k^{\beta_k}

Nvmervs natvralis dd nvmervm aa dividit tvm demvm cvm in eivs resolvtione nvlli alii primi apparent qvam ii nvmeri aa, et qvisqve exponente non maiore qvam eo qvem in aa habet. Inde seqvitvr dd simvl aa et bb dividere cvm exponens cvivsqve primi in dd nevtrvm e dvobvs exponentibvs svperat; maximvs ergo commvnis divisor obtinetvr svmendo pro qvoqve primo minorem e dvobvs exponentibvs. Eodem modo mm mvltiplvm commvne est cvm exponens cvivsqve primi in mm ambos attingit, et minimvm commvne mvltiplvm pro qvoqve primo maiorem e dvobvs exponentibvs svmit. Exempli gratia:

360=23⋅32⋅51⋅70,84=22⋅31⋅50⋅71360 = 2^3 \cdot 3^2 \cdot 5^1 \cdot 7^0, \qquad 84 = 2^2 \cdot 3^1 \cdot 5^0 \cdot 7^1
mcd⁡(360,84)=22⋅31⋅50⋅70=12,mcm⁡(360,84)=23⋅32⋅51⋅71=2520\operatorname{mcd}(360, 84) = 2^2 \cdot 3^1 \cdot 5^0 \cdot 7^0 = 12, \qquad \operatorname{mcm}(360, 84) = 2^3 \cdot 3^2 \cdot 5^1 \cdot 7^1 = 2520

Methodvs clara est, sed ab arte factores qvaerendi pendet, et magnorvm nvmerorvm factores qvaerere magni laboris est: nvllvs modvs celer ad id faciendvm notvs est. Algorithmvs sectionis seqventis maximvm commvnem divisorem sine factoribvs qvaerendis compvtat, solis divisionibvs.