If a and b are distinct integers, prove that a – b is a factor of an – bn, whenever n is a positive integer. [Hint write an = (a – b + b)n and expand]

Asked by Pragya Singh | 1 year ago |  128

Solution :-

In order to prove that (a – b) is a factor of (an – bn), it has to be proved that

an – bn = k (a – b) where k is some natural number.

a can be written as a = a – b + b

an = (a – b + b)= [(a – b) + b]n

nC0 (a – b)n + nC1 (a – b)n-1 b + …… + n bn

an – bn = (a – b) [(a –b)n-1 + nC1 (a – b)n-1 b + …… + n bn]

an – bn = (a – b) k

Where k = [(a –b)n-1 + nC1 (a – b)n-1 b + …… + n bn] is a natural number

This shows that (a – b) is a factor of (an – bn), where n is positive integer.

Answered by Abhisek | 1 year ago

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