<p>How many pairs of positive integers m, n satisfy \(\dfrac{1}{m} + \dfrac{4}{n} = \dfrac{1}{12}\) where n is an odd integer less than 60?</p>
Step-by-Step Solution
Key Concept: Rearrange the equation to isolate one variable, then use divisibility conditions to find when both m and n are positive integers. The constraint that n is odd restricts which solutions are valid.
<p><strong>Step 1:</strong> Rearrange the equation <span>1/m + 4/n = 1/12</span></p><p>Multiply through by 12mn: <span>12n + 48m = mn</span></p><p>Rearrange: <span>mn - 48m - 12n = 0</span></p><p><strong>Step 2:</strong> Add 576 to both sides to factor: <span>mn - 48m - 12n + 576 = 576</span></p><p><span>m(n - 48) - 12(n - 48) = 576</span></p><p><span>(m - 12)(n - 48) = 576</span></p><p><strong>Step 3:</strong> Let n - 48 = d where d is a divisor of 576. Then m - 12 = 576/d, so:</p><p><span>n = 48 + d</span> and <span>m = 12 + 576/d</span></p><p><strong>Step 4:</strong> For m, n to be positive integers: d must be a positive divisor of 576.</p><p>For n to be odd: <span>48 + d</span> must be odd. Since 48 is even, d must be odd.</p><p><strong>Step 5:</strong> Find odd divisors of 576 = 2⁶ × 3²:</p><p>Odd divisors are divisors of 3² = 9: {1, 3, 9}</p><p><strong>Step 6:</strong> Check constraint n < 60:</p><p>• d = 1: n = 49 ✓ (odd, < 60), m = 588</p><p>• d = 3: n = 51 ✓ (odd, < 60), m = 204</p><p>• d = 9: n = 57 ✓ (odd, < 60), m = 76</p><p><strong>Step 7:</strong> Verify one solution (d = 1): 1/588 + 4/49 = 1/588 + 48/588 = 49/588 = 1/12 ✓</p><p>∴ Answer: <strong>3 pairs</strong> (A)</p>
Correct Answer: A