Answered

Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Our Q&A platform offers a seamless experience for finding reliable answers from experts in various disciplines. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.

prove that x^n-Y^n divisible by x-y for all natural numbers x,y (x!=y),and n.

Sagot :

Let's do that by induction :
For [tex]n=1[/tex], [tex]x^1-y^1[/tex] is obviously divisible by [tex]x-y[/tex]

If we assume the property holds at rank [tex]n[/tex], then [tex]x^{n+1}-y^{n+1}=x(x^n-y^n)+y^n(x-y)[/tex]. Since [tex]x^n-y^n[/tex] is divisible by [tex](x-y)[/tex], we have [tex]A[/tex] such that [tex]x^n-y^n=A(x-y)[/tex]  hence [tex]x^{n+1}-y^{n+1}=(x-y)(Ax+y^n)[/tex].

Hence by induction for all [tex]n\ge1[/tex], [tex]x-y[/tex] divides [tex]x^n-y^n[/tex]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.