Question:
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]
Answer:
Let a = a + b - b
=> a = (a- b) + b
Now take power of n on both side
an = {(a-b) + b}n
= bn + nC1 *bn-1 *(a-b) + nC2 *bn-2 *(a-b)2 +........... + nCn *(a-b)n
=> an - bn = nC1 *bn-1 *(a-b) + nC2 *bn-2 *(a-b)2 +........... + nCn *(a-b)n
=> an - bn = (a-b)*{nC1 *bn-1 + nC2 *bn-2 *(a-b)1 +........... + nCn *(a-b)n-1 }
So (a-b) is a factor of an - bn where n is apositive integer.
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