<p>Let m be a positive integer and let the lines \(13x + 11y = 700\) and \(y = mx - 1\) intersect in a point whose coordinates are integers. Then m equals to:</p>
Step-by-Step Solution
Key Concept: For integer coordinates, the denominator must divide the numerator evenly; find factors of the constant term
<p><strong>Solution:</strong> Substitute \(y = mx - 1\) into \(13x + 11y = 700\):</p><p>\[13x + 11(mx - 1) = 700\]</p><p>\[13x + 11mx - 11 = 700\]</p><p>\[x(13 + 11m) = 711\]</p><p>\[x = \frac{711}{13 + 11m}\]</p><p>For x to be an integer, \((13 + 11m)\) must divide 711. Finding factors of 711 = 9 × 79:</p><p>If \(13 + 11m = 79\), then \(11m = 66\), so \(m = 6\).</p><p>However, checking with \(m = 5\): \(13 + 55 = 68\), which doesn't divide 711 evenly.</p><p>Rechecking: 711 = 3² × 79. Testing m=5 gives \(13 + 55 = 68\). Actually, \(13 + 11(5) = 68\). Then \(x = 711/68\) which is not an integer.</p><p>Testing \(m = 6\): \(13 + 66 = 79\), and \(711/79 = 9\) ✓. Then \(y = 6(9) - 1 = 53\).</p><p>∴ Answer is (c) 6.</p>
Correct Answer: b