If $a$ and $b$ are distinct integers, prove that $a-b$ is a factor of $a^n-b^n$, whenever $n$ is a positive integer.

*Hint: write $a^n=(a-b+b)^n$ and expand.*

Question

If $a$ and $b$ are distinct integers, prove that $a-b$ is a factor of $a^n-b^n$, whenever $n$ is a positive integer.

*Hint: write $a^n=(a-b+b)^n$ and expand.*

Accepted Answer

> [!TLDR]
>
> Expand $a^n=\big((a-b)+b\big)^n$ by the binomial theorem; the $k=0$ term is exactly $b^n$, and every other term visibly contains a factor of $(a-b)$ — factoring that out proves the divisibility.

**Step 1. Write $a=(a-b)+b$ and apply the binomial theorem to $a^n$.**
$$a^n = \big((a-b)+b\big)^n = \sum_{k=0}^n {}^nC_k\,(a-b)^k\,b^{n-k}.$$

**Step 2. Separate the $k=0$ term.** The $k=0$ term is ${}^nC_0(a-b)^0b^n=b^n$. So
$$a^n - b^n = \sum_{k=1}^n {}^nC_k\,(a-b)^k\,b^{n-k}.$$

**Step 3. Factor $(a-b)$ out of every remaining term.** Each term in the sum ($k\ge1$) carries at least one power of $(a-b)$:
$$a^n-b^n = (a-b)\left[\sum_{k=1}^n {}^nC_k\,(a-b)^{k-1}\,b^{n-k}\right].$$ …

Power	Expansion
$(x + y)^0$	$1$
$(x + y)^1$	$x + y$
$(x + y)^2$	$x^2 + 2xy + y^2$
$(x + y)^3$	$x^3 + 3x^2y + 3xy^2 + y^3$
$(x + y)^4$	$x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4$

Q.If $a$ and $b$ are distinct integers, prove that $a-b$ is a factor of $a^n-b^n$ , whenever $n$ is a positive integer. Hint: write $a^n=(a-b+b)^n$ and expand.

Concept understanding — Binomial Theorem

The Binomial Theorem: From Patterns to Power

The Pattern You Already Know

Why Do These Coefficients Appear?

The Precise Statement

A Quick Example

Q.If aaa and bbb are distinct integers, prove that a−ba-ba−b is a factor of an−bna^n-b^nan−bn, whenever nnn is a positive integer. Hint: write an=(a−b+b)na^n=(a-b+b)^nan=(a−b+b)n and expand.

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Q.If $a$ and $b$ are distinct integers, prove that $a-b$ is a factor of $a^n-b^n$ , whenever $n$ is a positive integer. Hint: write $a^n=(a-b+b)^n$ and expand.

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