Question:** What is the remainder when \( 1001 + 1003 + 1005 + 1007 + 1009 \) is divided by 11?
["Question: What is the remainder when ( 1001 + 1003 + 1005 + 1007 + 1009 ) is divided by 11?", "---", "Understanding the Remainder When Dividing a Sum by 11", "When faced with a math problem involving large numbers like ( 1001 + 1003 + 1005 + 1007 + 1009 ) divided by 11, a powerful strategy is to simplify using modular arithmetic—specifically, computing the sum modulo 11. This approach breaks down complex calculations into manageable steps, making it easier to determine the final remainder without performing massive divisions.", "This article explores how to find the remainder when the sum of five consecutive odd numbers—1001, 1003, 1005, 1007, and 1009—is divided by 11. We'll walk through the process step-by-step and highlight why modular arithmetic simplifies this kind of problem.", "---", "## Why Use Modular Arithmetic?", "The modulo operation finds the remainder after division by a number—in this case, 11. When adding multiple numbers, instead of computing the full sum first, computing each number modulo 11 and then adding those remainders makes the task faster and less error-prone.", "### Key Idea:\n[\n(a + b + c + d + e) \mod 11 \equiv [(a \mod 11) + (b \mod 11) + (c \mod 11) + (d \mod 11) + (e \mod 11)] \mod 11\n]", "---", "## Step-by-Step Calculation", "### Step 1: Find the remainder of each number modulo 11", "We compute ( 1001 \mod 11 ), ( 1003 \mod 11 ), and so on, using division and remainders.", "#### For 1001:\nDivide 1001 by 11:\n[\n1001 \div 11 = 91 \quad \ ext{(exact division, so remainder is 0)}\n]\n[\n1001 \equiv 0 \mod 11\n]", "#### For 1003:\n[\n1003 \div 11 = 91 \ ext{ remainder } 2 \quad (\ ext{since } 11 \ imes 91 = 1001, ; 1003 - 1001 = 2)\n]\n[\n1003 \equiv 2 \mod 11\n]", "#### For 1005:\n[\n1005 - 1001 = 4 \implies 1005 \equiv 4 \mod 11\n]", "#### For 1007:\n[\n1007 - 1001 = 6 \implies 1007 \equiv 6 \mod 11\n]", "#### For 1009:\n[\n1009 - 1001 = 8 \implies 1009 \equiv 8 \mod 11\n]", "---", "### Step 2: Add the remainders", "Now sum the computed remainders:", "[\n0 + 2 + 4 + 6 + 8 = 20\n]", "Now find ( 20 \mod 11 ):", "[\n20 \div 11 = 1 \ ext{ remainder } 9\n]\nSo,\n[\n20 \equiv 9 \mod 11\n]", "---", "### Step 3: Conclusion", "The remainder when ( 1001 + 1003 + 1005 + 1007 + 1009 ) is divided by 11 is 9.", "[\n\boxed{9}\n]", "---", "## Why This Method Works", "Instead of adding up the five five-digit numbers (which gives 5025) and then dividing 5025 by 11—a process prone to arithmetic errors—breaking the problem into modulo components reduces complexity and improves computational accuracy.", "---", "## Final Thoughts", "When asked, “What is the remainder when ( 1001 + 1003 + 1005 + 1007 + 1009 ) is divided by 11?” the insightful approach lies in modular arithmetic. By computing each term mod 11 and summing those remainders, we find the remainder efficiently and reliably. This method is especially useful in competitive math and programming challenges where speed and precision are essential.", "---", "Keywords: remainder when divided by 11, modular arithmetic, sum modulo 11, math problem solution, 1001 + 1003 + 1005 + 1007 + 1009, division remainder calculation, number theory, quick math tip", "Meta Description: Learn how to find the remainder when ( 1001 + 1003 + 1005 + 1007 + 1009 ) is divided by 11 using modular arithmetic. Step-by-step breakdown for accurate and fast computation."]
