# Why $a^n - b^n$ is divisible by $a-b$?

I did some mathematical induction problems on divisibility

• $9^n$ $-$ $2^n$ is divisible by 7.
• $4^n$ $-$ $1$ is divisible by 3.
• $9^n$ $-$ $4^n$ is divisible by 5.

Can these be generalized as $a^n$ $-$ $b^n$$= (a-b)N, where N is an integer? But why is a^n - b^n$$ = (a-b)N$ ?

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