File:GeoplanPpcm.png

Summary[edit]

Une interprétation graphique d'un système de congruences, avec ${\displaystyle a,b,r,s}$ donnés, trouver ${\displaystyle x}$ tel que

${\displaystyle \left\{{\begin{array}{l}x\equiv r[a]\\x\equiv s[b]\end{array}}\right.}$

Ce problème est expliqué simplement avec des engrenages: une roue dentée comportant ${\displaystyle a}$ dents s'engrène dans une autre roue similaire de ${\displaystyle b}$ dents (et dans une tringle horizontale, pour l'abscisse). Combien de dents doivent passer pour que sa ${\displaystyle r}$-ième dent vienne en coïncidence avec la ${\displaystyle s}$-ième dent ? Le PPCM des deux nombres ${\displaystyle a}$ et ${\displaystyle b}$ est ce qui permet de comprendre le comportement périodique de ce système, c'est le nombre de dents séparant deux contacts de même congruence. On peut donc trouver la solution s'il y en a une, dans l'intervalle ${\displaystyle [1,PPCM(a,b)]}$. Il y a une solution si ${\displaystyle {\rm {{PGCD}(a,b)|r-s}}}$.

Dans l'exemple en image, ${\displaystyle \left\{{\begin{array}{l}x\equiv 4[15]\\x\equiv 10[12]\end{array}}\right.}$ PPCM(12,15)=60 , pour x ∈ [1,60] , la solution est x = 34 .

Liens[edit]

Une page de géométrie interactive Geoplan (internet Explorer 6 avec les ActiveX installés).

Licensing[edit]

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