CBSE Class 11-science Answered
prove
Asked by Kirandoon | 26 Aug, 2008, 10:50: PM
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 | 12 Sep, 2008, 09:25: PM
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