<p><strong>For Problems 1–3</strong><br>Sum of certain consecutive odd positive integers is \(57^2 - 13^2\).</p><p><strong>Problem 3:</strong> The greatest integer is</p>
Step-by-Step Solution
Key Concept: Use the difference of squares formula to factor 57² - 13² = (57+13)(57-13) = 70×44, then recognize this as the sum of consecutive odd integers and apply the formula: sum of n consecutive odd integers starting from (2a+1) equals n(2a+n), which gives n² + 2an = sum.
<p><strong>Step 1:</strong> Calculate 57² - 13² using difference of squares.</p><p>57² - 13² = (57+13)(57-13) = 70 × 44 = 3080</p><p><strong>Step 2:</strong> For n consecutive odd integers starting from (2a+1), the sum is n(2a+n) where a ≥ 0.</p><p>Set n(2a+n) = 3080, so 2an + n² = 3080, giving a = (3080-n²)/(2n)</p><p><strong>Step 3:</strong> For a to be a non-negative integer, n must be a divisor of 3080 = 2³ × 5 × 7 × 11, and (3080-n²) ≥ 0.</p><p><strong>Step 4:</strong> Test divisors: For n = 8: a = (3080-64)/16 = 3016/16 = 188.5 ✗</p><p>For n = 4: a = (3080-16)/8 = 3064/8 = 383 ✓</p><p>For n = 70: a = (3080-4900)/140 = negative ✗</p><p><strong>Step 5:</strong> With n = 4 and a = 383, the first odd integer is 2(383)+1 = 767.</p><p>The four consecutive odd integers are: 767, 769, 771, 773</p><p>Verify: 767 + 769 + 771 + 773 = 3080 ✓</p><p><strong>Step 6:</strong> The greatest integer is 773.</p><p>∴ Answer: B</p>
Correct Answer: B