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]

Question

TextProof of Binomial Theorem by Mathematical Induction

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Using binomial theorem, prove that 6ⁿ–5n always leaves remainder 1 when divided by 25.

Show that 9n+1 – 8n – 9 is divisible by 64, whenever n is a positive integer.