Let a,b be natural numbers. Prove that a^n-b^n is divisible by a-b for all n whe

Question

Let a,b be natural numbers. Prove that a^n-b^n is divisible by a-b for all n where n is a natural number.

So far this is what i have.

We will prove this using incution on n. First, suppose that n=1. Then a^1-b^1=a-b. So a^1-b^1 is in fact divisble by a-b. Therefore it holds for n=1. Suppose that a^n-b^n is divisible by a-b which means a^n-b^n=(a-b)k for some k. To who that a^(n+1)-b^(n+1) is divisible by a-b we will make suse of our hypothesis that a^n-b^n is divisble by a-b.

a^(n+1)-b^(n+1)=a*^n-b*b^n=?

This is all I have. I need a full and detailed proof using mathmatical induction. Thankyou

Explanation / Answer

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

This is divisible by (a - b) thanks to the induction step.

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