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Divisibilitas et divisio cvm residvo

Svmmarivm

Haec lectio relationem divisibilitatis ad nvmeros integros extendit eivsqve proprietates dedvcit: reflexivitatem, transitivitatem, divisibilitatem combinationvm linearivm et margines divisorvm. Deinde, principio bonae ordinationis adhibito, theorema divisionis cvm residvo demonstrat, qvod cviqve integro qvotientem et residvvm vnicos tribvit, idqve adhibet vt integri secvndvm residvvm distribvantvr et vt de svmmis, prodvctis et potentiis per casvs ratiocinetvr. Desinit in criteriis divisibilitatis per 22, 44, 55, 33, 99 et 1111, ex scriptvra decimali nvmerorvm demonstratis.

Proposita discendi

  1. Definitionem divisibilitatis in nvmeris integris eivsqve proprietates adhibere, praesertim divisibilitatem combinationvm linearivm.
  2. Exsistentiam et vnicitatem qvotientis et residvi demonstrare, eosqve etiam pro dividendis negativis compvtare.
  3. Integros secvndvm residvvm distribvere eaqve distribvtione vti ad proprietates per casvs demonstrandas.
  4. Criteria divisibilitatis per 22, 44, 55, 33, 99 et 1111 ex scriptvra decimali demonstrare et adhibere.

Divisibilitas in nvmeris integris

In lectione de operationibvs nvmerorvm natvralivm dictvm est nvmervm natvralem bb alivm nvmervm natvralem aa dividere, et scriptvm est b∣ab \mid a, cvm aa mvltiplvm sit ipsivs bb. Definitio sine vlla mvtatione ad nvmeros integros extenditvr, qvorvm constrvctio in lectione de nvmeris integris tractatvr: integer bb integrvm aa dividit si aa prodvctvm est ipsivs bb et alicvivs integri. Ea lectio ad partem seqventem pertinet; hic integri anticipantvr, et regvlae eorvm arithmeticae ordinisqve qvae ibi fvndantvr sine demonstratione admittvntvr: regvla signorvm, prodvctvm integri et nvmeri natvralis sensvm inaeqvalitatvm servare, factorem a nihilo diversvm in prodvcto cancellari posse, integros esse nvmeros natvrales, 00 et opposita nvmerorvm natvralivm. Earvm demonstratio constrvctionem integrorvm ex nvmeris natvralibvs postvlat, qvae nihil ex hac lectione adhibet, ita vt nvllvs circvlvs sit.

b∣a↔(∃k∈Z)(a=bk)b \mid a \leftrightarrow (\exists k \in \mathbb{Z})(a = bk)

Dicitvr etiam bb esse divisorem ipsivs aa et aa esse mvltiplvm ipsivs bb. Ex definitione statim casvs qvidam extremi seqvvntvr: 00 mvltiplvm est omnis integri, qvia 0=b⋅00 = b \cdot 0; nvmeri 11 et −1-1 omnem integrvm dividvnt; et 00 solvm 00 dividit, qvia 0⋅k=00 \cdot k = 0 pro omni kk. Atqvi a=bka = bk aeqvivalet ipsis −a=b(−k)-a = b(-k) et a=(−b)(−k)a = (-b)(-k); vnde aa et −a-a eosdem divisores habent, et bb et −b-b eosdem nvmeros dividvnt. Qva de cavsa in vsv satis est divisores natvrales considerare.

Proprietates divisibilitatis

Pro omnibvs integris aa, bb, cc, dd, xx et yy proprietates seqventes valent, qvarvm demonstrationes ad definitionem redigvntvr:

  1. Reflexivitas: a∣aa \mid a, qvia a=a⋅1a = a \cdot 1.
  2. Transitivitas: si a∣ba \mid b et b∣cb \mid c, tvm a∣ca \mid c. Etenim, si b=akb = ak et c=blc = bl, tvm c=a(kl)c = a(kl).
  3. Combinationes lineares: si d∣ad \mid a et d∣bd \mid b, tvm d∣ax+byd \mid ax + by, vt propositio seqvens demonstrat.
  4. Margines: si b∣ab \mid a et a>0a > 0, tvm −a≤b≤a-a \leq b \leq a. Satis est id ostendere pro b>0b > 0: tvm a=bka = bk cvm kk natvrali, et qvoniam k≥1k \geq 1, habemvs a=b+b(k−1)≥ba = b + b(k - 1) \geq b.
  5. Antisymmetria in nvmeris natvralibvs: si aa et bb natvrales svnt, a∣ba \mid b et b∣ab \mid a, tvm a=ba = b, qvia margines dant a≤ba \leq b et b≤ab \leq a.

Margines conseqventiam habent qva saepe vtemvr: integer a nihilo diversvs finitvm tantvm divisorvm nvmervm habet. Ex omnibvs proprietatibvs tertia fecvndissima est, eamqve cvm envntiatione integra et demonstratione memoria tenere expedit.

PropositioCombinationes lineares

Si integer dvos alios dividit, omnem qvoqve eorvm combinationem linearem cvm coefficientibvs integris dividit. Praesertim eorvm svmmam, differentiam et qvodlibet mvltiplvm vtrivsqve dividit. Figvra demonstrationem comitatvr cvm d=3d = 3, a=6a = 6, b=9b = 9, x=2x = 2 et y=−1y = -1, et qvodqve mvltiplvm ipsivs 33 vt versvm acervorvm ternorvm depingit.

Demonstratio

  1. a=dk,b=dl,k,l∈Z\dato{k}{a = dk}, \quad \dato{l}{b = dl}, \quad k, l \in \mathbb{Z}

    Per definitionem divisibilitatis, vterqve integer prodvctvm est ipsivs dd et alicvivs integri.

  2. ax+by=dk x+dl y=d(kx+ly)ax + by = \resaltar{dk}\,x + \resaltar{dl}\,y = \resaltar{d(kx + ly)}

    In combinationem linearem svbstitvimvs et dd vt factorem commvnem extrahimvs, per distribvtivitatem.

  3. kx+ly∈Z→d∣ax+bykx + ly \in \mathbb{Z} \rightarrow d \mid ax + by

    Nvmervs kx+lykx + ly integer est, qvia svmma et prodvctvm integrorvm integri svnt; ergo combinatio mvltiplvm est ipsivs dd.

  4. d∣a+b,d∣a−b,d∣ax,d∣byd \mid a + b, \quad d \mid a - b, \quad d \mid ax, \quad d \mid by

    Cvm x=y=1x = y = 1 svmma obtinetvr; cvm x=1x = 1 et y=−1y = -1, differentia; cvm y=0y = 0, mvltipla ipsivs aa, et eodem modo ea ipsivs bb.

(d∣a∧d∣b)→d∣ax+by(d \mid a \wedge d \mid b) \rightarrow d \mid ax + by

Contrario sensv lecta, proprietas ad divisores exclvdendos valet: si dd dividit aa sed non bb, tvm non dividit a+ba + b, qvia, si divideret, etiam differentiam (a+b)−a=b(a + b) - a = b divideret.