Power Difference Divisible by x Minus y

Induction is one of the most powerful tools in HSC Extension 1 and 2, allowing us to prove statements for all positive integers. In this problem, we revisit a classic algebraic result: the difference of two powers can be factored out by the difference of their bases. By carefully adding and subtracting a cross term, you’ll learn how to connect the inductive hypothesis to the target expression — a technique that appears in polynomial factorisation, complex numbers, and beyond.

Problem Statement

Prove by induction that $x^n - y^n$ is divisible by $x - y$ for all positive integers $n$, where $x, y$ are integers with $x \neq y$.

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Hints

Attempt the proof independently first. Focus on the key theorem, algebraic transformation, or contradiction setup that links the hypothesis to the target conclusion.

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Solutions

Proof by Mathematical Induction

We start by checking the simplest case, $n=1$, which is trivially true.

Base Case ($n=1$):

For $n=1$:

$$x^1 - y^1 = x - y$$

Since $x - y$ is divisible by $x - y$, the statement holds for $n=1$. \checkmark

Next, we assume the statement holds for some arbitrary positive integer $k$, giving us a factorisation we can use later.

Inductive Hypothesis:

Assume the statement is true for $n=k$, where $k$ is a positive integer:

$$x^k - y^k \text{ is divisible by } x - y$$

This means we can write:

$$x^k - y^k = m(x - y)$$

for some integer $m$.

Now the key is to write $x^{k+1} - y^{k+1}$ in a form that reveals the inductive hypothesis. We do this by adding and subtracting $xy^k$: from the first pair we can factor out $x$, and from the second $y^k$, exposing the known factors.

Inductive Step:

We must prove the statement for $n=k+1$:

$$x^{k+1} - y^{k+1} \text{ is divisible by } x - y$$

Start with $x^{k+1} - y^{k+1}$ and manipulate to introduce $x^k - y^k$:

$$\begin{aligned} x^{k+1} - y^{k+1} &= x^{k+1} - xy^k + xy^k - y^{k+1} \\ &= x(x^k - y^k) + y^k(x - y) \end{aligned}$$

Substitute the inductive hypothesis $x^k - y^k = m(x-y)$:

$$\begin{aligned} x^{k+1} - y^{k+1} &= x \cdot m(x-y) + y^k(x-y) \\ &= (x-y)(xm + y^k) \end{aligned}$$

Since $x, y, m, k$ are all integers, $M = xm + y^k$ is an integer.

Therefore:

$$x^{k+1} - y^{k+1} = M(x-y)$$

This shows $x^{k+1} - y^{k+1}$ is divisible by $x-y$. \checkmark

Conclusion:

By mathematical induction, $x^n - y^n$ is divisible by $x - y$ for all positive integers $n$.

Note: This problem demonstrates the crossover between Induction and Proofs topics in HSC Extension 2.

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Takeaways

• Strategic Addition/Subtraction: Add and subtract $xy^k$ to create terms involving $x^k - y^k$ and $x - y$
• Factorization: Extract common factor $(x-y)$ after rearrangement
• Induction Crossover: Problem appears in both HSC-Induction and HSC-Proofs collections, showing technique overlap
• Alternative Formula: Result leads to $x^n - y^n = (x-y)(x^{n-1} + x^{n-2}y + \ldots + xy^{n-2} + y^{n-1})$

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Further Readings

If you found this proof interesting, be sure to check out these relevant HSC booklets to sharpen your reasoning skills: HSC Proofs, HSC Complex Numbers, HSC Polys Ext 1