if a and b are two distinct integers , prove that a-b is a factor of a^n - b^n
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Answer:
For n=1,a
n
−b
n
=a−b
Let for n=k,a
k
−b
k
=λ(a−b)
for n=k+1,a
n
−b
n
=a
k+1
−b
k+1
=a.a
k
−b.b
k
=a[b
k
+λ(a−b)]−b
k+1
=ab
k
−b
k+1
+λa(a−b)
=b
k
(a−b)+λa(a−b)
=(a−b)(b
k
+λa)
=λ
′
(a−b)
⇒ By induction a
n
−b
n
is always divisible by a−b
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