(x^n − a^n) is divisible by (x − a) when n equals…?

Given data

• Divisibility of x^n − a^n by x − a.

Concept / Approach

• Factorization identity: x^n − a^n = (x − a)(x^{n−1} + x^{n−2}a + ⋯ + xa^{n−2} + a^{n−1})
• The identity holds for all integers n ≥ 1 (natural numbers).

Step-by-step reasoning

Since x^n − a^n has (x − a) as a factor for any n ≥ 1, the quotient is the (n−1)-term geometric sum shown above.

Verification

Check n = 1: x − a is trivially divisible by x − a.Check n = 2: x^2 − a^2 = (x − a)(x + a).

Common pitfalls

• Thinking it holds only for even or odd n; in fact it holds for every positive integer n.

Final Answer

Any positive integer n.