Question: What is the remainder when $ 11071 + 11073 + 11075 + 11077 $ is divided by $ 8 $?
["SEO-Optimized Article: Find the Remainder of $ 11071 + 11073 + 11075 + 11077 $ Divided by 8", "When faced with a math problem like "What is the remainder when $11071 + 11073 + 11075 + 11077$ is divided by 8?", it might seem complex at first glance — but with a few strategic simplifications, even number theory questions become approachable. In this article, we break down how to efficiently calculate the remainder (also known as the modulo result) without adding the large numbers first, making the process fast, clear, and ideal for students and math enthusiasts alike.", "---", "## Understanding the Problem", "We are asked to compute:\n$$\n(11071 + 11073 + 11075 + 11077) \mod 8\n$$", "Directly adding the four odd numbers and dividing by 8 seems tedious. However, modular arithmetic allows us to simplify each number modulo 8 before summing, due to the properties of remainders.", "---", "## Step 1: Use Modular Arithmetic to Reduce Each Number", "Instead of adding first, compute each number mod 8:", "$$\n11071 \mod 8,\quad 11073 \mod 8,\quad 11075 \mod 8,\quad 11077 \mod 8\n$$", "To find $ n \mod 8 $, focus on the last three digits of $ n $, since $ 1000 \div 8 = 125 $, so only the last three digits determine the remainder.", "- $ 11071 \rightarrow 071 = 71 $\n- $ 11073 \rightarrow 073 = 73 $\n- $ 11075 \rightarrow 075 = 75 $\n- $ 11077 \rightarrow 077 = 77 $", "Now compute each modulo 8:", "### Compute $ 71 \mod 8 $", "$$\n71 \div 8 = 8 \ imes 8 = 64,\quad 71 - 64 = 7 \Rightarrow 71 \equiv 7 \mod 8\n$$", "### Compute $ 73 \mod 8 $", "$$\n73 - 64 = 9,\quad 9 - 8 = 1 \Rightarrow 73 \equiv 1 \mod 8\n$$", "### Compute $ 75 \mod 8 $", "$$\n75 - 72 = 3 \quad (\ ext{since } 8 \ imes 9 = 72) \Rightarrow 75 \equiv 3 \mod 8\n$$", "### Compute $ 77 \mod 8 $", "$$\n77 - 72 = 5 \Rightarrow 77 \equiv 5 \mod 8\n$$", "---", "## Step 2: Sum the Remainders Modulo 8", "Now sum the residues and reduce the total modulo 8:", "$$\n7 + 1 + 3 + 5 = 16\n$$", "$$\n16 \mod 8 = 0 \quad (\ ext{since } 8 \ imes 2 = 16)\n$$", "---", "## Final Answer", "$$\n(11071 + 11073 + 11075 + 11077) \mod 8 = \boxed{0}\n$$", "---", "## Why This Method Works: Key Insights", "- Modulo Properties: $ (a + b) \mod m = [(a \mod m) + (b \mod m)] \mod m $\n- Efficiency: Avoiding large additions and instead reducing each term first saves time and reduces error.\n- Pattern Recognition: Noticing that the numbers form an arithmetic sequence with common difference 2 helps confirm expected modulo behavior.", "---", "## Practical Usage: Why This Matters", "Understanding how to compute remainders efficiently is essential in:", "- Computer science (e.g., hashing, cyclic computations)\n- Cryptography (modular exponentiation)\n- Competitive programming and math exams", "---", "## More Explanation: Quick Reference for Last Three Digits", "For any odd number, take the last three digits:", "| Number | Last Three Digits | $ \mod 8 $ |\n|---------|------------------|------------|\n| 071 | 71 | 7 |\n| 073 | 73 | 1 |\n| 075 | 075 | 3 |\n| 077 | 077 | 5 |\n| Sum | — | 16 → 0 |", "---", "## Summary", "To find the remainder when $11071 + 11073 + 11075 + 11077$ is divided by 8:", "1. Compute each number modulo 8 using last three digits.\n2. Sum the remainders: $7 + 1 + 3 + 5 = 16$.\n3. Take $16 \mod 8 = 0$.", "Answer: 0", "Mastering modular arithmetic turns complex problems into simple steps — perfect for faster, smarter problem-solving.", "---", "Keywords: remainder when divided by 8, modular arithmetic, sum modulo 8, 11071 + 11073 + 11075 + 11077 mod 8, math quick problem solution, modular properties, last three digits method, computer science math, competitive math tips.", "---", "Optimization Tip: Remember — for sums or sequences mod $ m $, identifying patterns and reducing terms mod $ m $ first drastically simplifies calculations. Apply this trick next time to problems like this!"]
