<p>The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is __________.</p>
Step-by-Step Solution
Key Concept: Partition two-digit numbers into two arithmetic progressions based on remainder conditions (remainder 2: 16,23,30,...,93 and remainder 5: 12,19,26,...,96), then sum each AP separately using the formula S = n(first+last)/2.
<p><strong>Step 1:</strong> Find two-digit numbers with remainder 2 when divided by 7.</p><p>Form: 7k + 2, where 10 ≤ 7k + 2 ≤ 99</p><p>8 ≤ 7k ≤ 97 → k ∈ {2,3,4,...,13}</p><p>Numbers: 16, 23, 30, 37, 44, 51, 58, 65, 72, 79, 86, 93 (12 terms)</p><p>Sum₁ = 12(16 + 93)/2 = 12 × 109/2 = 654</p><p><strong>Step 2:</strong> Find two-digit numbers with remainder 5 when divided by 7.</p><p>Form: 7m + 5, where 10 ≤ 7m + 5 ≤ 99</p><p>5 ≤ 7m ≤ 94 → m ∈ {1,2,3,...,13}</p><p>Numbers: 12, 19, 26, 33, 40, 47, 54, 61, 68, 75, 82, 89, 96 (13 terms)</p><p>Sum₂ = 13(12 + 96)/2 = 13 × 108/2 = 702</p><p><strong>Step 3:</strong> Total sum = Sum₁ + Sum₂ = 654 + 702 = 1356</p><p>∴ Answer: <strong>1356</strong></p>
Correct Answer: 1256