<p>Two numbers \(a, b\) are chosen from the set of integers 1, 2, 3, …, 39. Then probability that the equation \(7a - 9b = 0\) is satisfied is</p>
Step-by-Step Solution
Key Concept: For 7a - 9b = 0 to hold, we need 7a = 9b. Since gcd(7,9) = 1, we must have a = 9k and b = 7k for positive integer k. Count valid pairs (a,b) within [1,39] and divide by total ways to choose 2 numbers.
<p><strong>Step 1:</strong> From 7a - 9b = 0, we get 7a = 9b, so a/b = 9/7.</p><p><strong>Step 2:</strong> Since gcd(7,9) = 1, this means a = 9k and b = 7k for some positive integer k.</p><p><strong>Step 3:</strong> For both a and b to lie in {1, 2, 3, …, 39}:</p><p>• 1 ≤ 9k ≤ 39 ⟹ k ≤ 4.33... ⟹ k ∈ {1, 2, 3, 4}</p><p>• 1 ≤ 7k ≤ 39 ⟹ k ≤ 5.57... ⟹ k ∈ {1, 2, 3, 4, 5}</p><p>Common values: k ∈ {1, 2, 3, 4}</p><p><strong>Step 4:</strong> Valid pairs (a,b):</p><p>• k = 1: (9, 7)</p><p>• k = 2: (18, 14)</p><p>• k = 3: (27, 21)</p><p>• k = 4: (36, 28)</p><p>Total: 4 pairs</p><p><strong>Step 5:</strong> Total ways to choose 2 distinct numbers from 39: C(39,2) = (39 × 38)/2 = 741</p><p>∴ Probability = 4/741</p>
Correct Answer: A