Q4.    If a and b are distinct integers, prove thata - b is a factor of a^n - b^n , whenever n is a positive integer.
[Hint: write a^n = (a - b + b)^n and expand]

Answers (1)

we need to prove, 

a^n-b^n=k(a-b)  where k is some natural number.

Now let's add and subtract b from a so that we can prove the above result,

a=a-b+b

a^n=(a-b+b)^n=[(a-b)+b]^n

=^nC_0(a-b)^n+^nC_1(a-b)^{n-1}b+........^nC_nb^n

=(a-b)^n+^nC_1(a-b)^{n-1}b+........^nC_{n-1}(a-b)b^{n-1}+b^n\Rightarrow a^n-b^n=(a-b)[(a-b)^{n-1}+^nC_2(a-b)^{n-2}+........+^nC_{n-1}b^{n-1}]

\Rightarrow a^n-b^n=k(a-b)

Hence,a - b is a factor of a^n - b^n.

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