<p><strong>For Problems 1–3</strong><br>Sum of certain consecutive odd positive integers is \(57^2 - 13^2\).</p><p><strong>Problem 2:</strong> The least value of an integer is</p>
Step-by-Step Solution
Key Concept: Use the algebraic identity a² - b² = (a+b)(a-b) to factor 57² - 13² = 70 × 44, then recognize that the sum of n consecutive odd integers starting from (2k+1) equals n(2k+n), allowing us to find all possible sequences and their minimum starting values.
<p><strong>Step 1:</strong> Factor using difference of squares: 57² - 13² = (57+13)(57-13) = 70 × 44 = 3080</p><p><strong>Step 2:</strong> For n consecutive odd integers starting from (2k+1): Sum = n(2k+1) + n(n-1) = n(2k+n)</p><p>So we need: n(2k+n) = 3080, where k ≥ 0 (to ensure positive integers) and n ≥ 1</p><p><strong>Step 3:</strong> This gives 2k + n = 3080/n, so k = (3080/n - n)/2</p><p>For k to be a non-negative integer: 3080/n - n ≥ 0 and 3080/n - n must be even</p><p><strong>Step 4:</strong> Find divisors of 3080 = 2³ × 5 × 7 × 11. Test each divisor n:</p><p>• n = 4: 2k + 4 = 770 → k = 383 → First odd integer = 2(383)+1 = 767</p><p>• n = 8: 2k + 8 = 385 → k = 188.5 (not integer)</p><p>• n = 10: 2k + 10 = 308 → k = 149 → First odd integer = 2(149)+1 = 299</p><p>• n = 20: 2k + 20 = 154 → k = 67 → First odd integer = 2(67)+1 = 135</p><p>• n = 28: 2k + 28 = 110 → k = 41 → First odd integer = 2(41)+1 = 83</p><p>• n = 44: 2k + 44 = 70 → k = 13 → First odd integer = 2(13)+1 = 27</p><p>• n = 55: 2k + 55 = 56 → k = 0.5 (not integer)</p><p>• n = 70: 2k + 70 = 44 → k = -13 (negative, invalid)</p><p><strong>Step 5:</strong> Valid sequences have minimum starting values: 27, 83, 135, 299, 767...</p><p>∴ Answer: B (The least value is 27)</p>
Correct Answer: B