<p>Find the minimum value of \(m + n\) where \(m\) and \(n\) are positive integers with \(m > 1\) satisfying:<br><br>\[\frac{m}{n} - \frac{1}{mn} = \frac{1}{2013}\]<br><br>and \(n = 2013m - \dfrac{2013}{m}\), with \(n\) being a positive integer.</p>
Step-by-Step Solution
Key Concept: Rearrange the given equation to m² - 2013m - 1 = 0, then recognize that n = 2013m - 2013/m forces m to divide 2013. Since 2013 = 3 × 11 × 61, test divisors to find which gives integer n and minimum m + n.
<p><strong>Step 1:</strong> Simplify the first equation. Multiply by mn: m² - 1 = mn/2013, so m² - mn/2013 - 1 = 0. Rearranging: m² - 2013m - 1 = 0 (after multiplying through by 2013, but we work with the constraint differently).</p><p><strong>Step 2:</strong> From m/n - 1/(mn) = 1/2013, multiply by mn: m² - 1 = n/2013. Thus n = 2013(m² - 1). But we're also given n = 2013m - 2013/m.</p><p><strong>Step 3:</strong> Equate the two expressions for n: 2013(m² - 1) = 2013m - 2013/m. Divide by 2013: m² - 1 = m - 1/m. Multiply by m: m³ - m = m² - 1, giving m³ - m² - m + 1 = 0, or (m-1)(m²-1) = 0... [Alternative approach] From n = 2013m - 2013/m, for n to be a positive integer, m must divide 2013.</p><p><strong>Step 4:</strong> Factor 2013 = 3 × 11 × 61. Divisors are: 1, 3, 11, 33, 61, 183, 671, 2013. Since m > 1, test m = 3: n = 2013(3) - 2013/3 = 6039 - 671 = 5368. Check: m + n = 3 + 5368 = 5371.</p><p><strong>Step 5:</strong> Test m = 11: n = 2013(11) - 2013/11 = 22143 - 183 = 21960. Then m + n = 11 + 21960 = 21971. Test m = 61: n = 2013(61) - 2013/61 = 122793 - 33 = 122760. Then m + n = 61 + 122760 = 122821. As m increases, m + n increases. Verify m = 3 satisfies the original equation.</p><p><strong>Step 6:</strong> For m = 3, n = 5368: m/n - 1/(mn) = 3/5368 - 1/16104 = 9/16104 - 1/16104 = 8/16104 = 1/2013. ✓</p><p>∴ Answer: 3 + 5368 = <strong>5371</strong></p>
Correct Answer: 371