<p>If \(x^n - 1\) is divisible by \(x - k\), then the least positive integral value of \(k\) is __________.</p>
Step-by-Step Solution
Key Concept: By the Factor Theorem, if (x - k) divides (x^n - 1), then x = k must be a root, meaning k^n = 1. The least positive integer k satisfying k^n = 1 for any positive integer n is k = 1.
<p><strong>Step 1:</strong> Apply the Factor Theorem. If (x - k) divides (x^n - 1), then (x^n - 1) must equal zero when x = k.</p><p><strong>Step 2:</strong> Substitute x = k: k^n - 1 = 0, which gives k^n = 1.</p><p><strong>Step 3:</strong> Find the least positive integer k such that k^n = 1 for any positive integer n.</p><p><strong>Step 4:</strong> The only positive integer satisfying k^n = 1 for all positive integers n is k = 1, since 1^n = 1 for every positive integer n.</p><p><strong>Step 5:</strong> Verify: (x^n - 1) is divisible by (x - 1) for all positive integers n, giving quotient x^(n-1) + x^(n-2) + ... + x + 1.</p><p>∴ Answer: 1</p>
Correct Answer: 1