Power Difference Divisible by x Minus y

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 $x \neq y$ .

Hints

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

Solutions

Proof by Mathematical Induction

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

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 $m$ .

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 $n$ .

\hfill $\square$

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

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 $x^n - y^n = (x-y)(x^{n-1} + x^{n-2}y + \ldots + xy^{n-2} + y^{n-1})$

Further Readings

If you found this proof interesting, be sure to check out these relevant HSC booklets to sharpen your reasoning skills:

HSC Proofs: https://vumaths.com/booklets/hsc-proofs/
HSC Complex Numbers: https://vumaths.com/booklets/hsc-complex-numbers/
HSC Polys Ext 1: https://vumaths.com/booklets/hsc-polys-ext-1/

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