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prove

Asked by Kirandoon 26th August 2008, 10:50 PM
Answered by Expert
Answer:

to prove that x-1 is a factor of x2n-1 - 1 , it is sufficient to prove that x2n-1 - 1 is divisible by x-1

we will prove it by mathematical induction

for n=1 , x-1 is divisible by x-1

for n=2 , x3-1 = (x-1)(x2+x+1) which is also divisible by x-1

let us assume that  x2n-1 - 1 is divisible by x-1 for all n N

now we have to show that it is also true for n= N+1

for n=N+1

x2N+1 - 1 = x2 x2N-1 - 1 + x2 -x2 = x2 (x2N-1 - 1) + (x2 - 1)

x2N-1 - 1 is divisible by x-1 ( true by our hypothesis)

and x2 - 1 = (x-1)(x+1) which is also divisible by x-1

so it is true for n= N+1

i.e. x2N-1 - 1 is divisible by x-1 or it has a factor equal to x-1

Answered by Expert 12th September 2008, 9:25 PM
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