Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

Question Number 162109 by SANOGO last updated on 26/Dec/21

Answered by TheHoneyCat last updated on 01/Jan/22

$$1)‘‘x/y^{″}\mathrm{signifie}‘‘x\mathrm{divise}{y}^{″}$$$$d=\mathrm{pgcd}(a,b)$$$$\mathrm{Donc}d/a\mathrm{et}d/b$$$$\mathrm{soit}\{\frac{a}{d},\frac{b}{d}\}\subset \mathbb{N}$$$$\mathrm{Ainsi}m=a\times \frac{b}{d}=b\times \frac{a}{d}\mathrm{est}\mathrm{trivialement}\mathrm{un}\mathrm{multiple}\mathrm{des}\mathrm{deux}\mathrm{entiers}$$$$$$$$2)\mathrm{soit}x=\mathrm{pgcd}(a_1,b_1)$$$$x/a_1\mathrm{et}x/b_1$$$${\mathrm{d}}^{\prime }\mathrm{ou}x.d/a\mathrm{et}x.d/b$$$$\mathrm{Or}d\mathrm{est}\mathrm{le}‘‘plus{grand}^{″}\mathrm{diviseur}\mathrm{commun}\mathrm{de}a\mathrm{et}b$$$$\mathrm{donc}\mathrm{pour}\mathrm{avoir}x.d⩽d\mathrm{il}\mathrm{faut}x=1$$$$\mathrm{ainsi}\mathrm{pdcg}(a_1,b_1)=1\mathrm{ils}\mathrm{sont}\mathrm{donc}\mathrm{premiers}\mathrm{entre}\mathrm{eux}...$$$$$$$$3)$$$$\mathrm{soit}x\mathrm{un}\mathrm{multiple}\mathrm{commun}\mathrm{de}a\mathrm{et}b$$$$a/x\mathrm{et}b/x$$$$\mathrm{donc}{a}_{1}d/x\mathrm{et}{b}_{1}d/x$$$$\mathrm{ou}{a}_1/\frac{x}{d}\mathrm{et}{b}_1/\frac{x}{d}$$$$a_1\mathrm{et}{b}_1\mathrm{etant}\mathrm{premiers}\mathrm{entre}\mathrm{eux},\mathrm{on}\mathrm{a}\mathrm{donc}$$$$a_1{b}_1/\frac{x}{d}\mathrm{ou}\mathrm{tout}\mathrm{simplement}{a}_1{b}_{1}d/x$$$$\mathrm{or}{a}_1{b}_{1}d=\frac{a}{d}\frac{b}{d}d=\frac{ab}{d}=m$$$${\mathrm{d}}^{\prime }\mathrm{ou}m/x$$$$x\mathrm{est}\mathrm{bien}\mathrm{multiple}\mathrm{de}\mathrm{m}{.}_{◼}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com