Question: What is the remainder when the sum $ 1^3 + 2^3 + 3^3 + \dots + 10^3 $ is divided by 7?
["SEO-Optimized Article: Finding the Remainder of $1^3 + 2^3 + \dots + 10^3$ Divided by 7", "---", "### What Is the Remainder When the Sum $1^3 + 2^3 + \dots + 10^3$ Is Divided by 7?", "When faced with the sum $1^3 + 2^3 + 3^3 + \dots + 10^3$, a common mathematical question arises: What is the remainder when this sum is divided by 7? This type of problem combines number theory with arithmetic sequences and modular arithmetic, making it a great example of applying basic number patterns in ways that are both practical and educational.", "---", "### The Formula for the Sum of Cubes", "Before calculating the remainder, it’s helpful to recall the well-known formula for the sum of the first $n$ cubes:", "$$\n1^3 + 2^3 + 3^3 + \dots + n^3 = \left( \frac{n(n+1)}{2} \right)^2\n$$", "For $n = 10$, the sum becomes:", "$$\n1^3 + 2^3 + \dots + 10^3 = \left( \frac{10 \cdot 11}{2} \right)^2 = (55)^2 = 3025\n$$", "Now we want to compute:", "$$\n3025 \mod 7\n$$", "---", "### Step-by-Step: Compute $3025 \div 7$ Remainder", "Rather than dividing 3025 by 7 directly, we simplify using modular arithmetic to make the process easier.", "1. First, find $55 \mod 7$:", "$$\n55 \div 7 = 7 \ ext{ remainder } 6 \quad \Rightarrow \quad 55 \equiv 6 \pmod{7}\n$$", "2. Then square it modulo 7:", "$$\n55^2 \equiv 6^2 = 36 \pmod{7}\n$$", "3. Now compute $36 \mod 7$:", "$$\n36 \div 7 = 5 \ ext{ remainder } 1 \quad \Rightarrow \quad 36 \equiv 1 \pmod{7}\n$$", "---", "### Conclusion", "Therefore, the remainder when $1^3 + 2^3 + \dots + 10^3$ is divided by 7 is:", "$$\n\boxed{1}\n$$", "This concise solution demonstrates how modular arithmetic can simplify seemingly complex sums—ideal for learners, students, and exam preparations in elementary number theory and algebra.", "---", "### Bonus: Practical Applications of This Concept", "Understanding how to compute large sums modulo small numbers is important in:", "- Cryptography, where modular arithmetic forms the backbone of many encryption algorithms.\n- Computer Science, especially in hashing and checksum functions.\n- Competitive Mathematics, where optimizing large computations uniquely separates strong problem-solvers.", "---", "Keywords: remainder when sum of cubes divided by 7, 1³ + 2³ + … + 10³ mod 7, number theory, modular arithmetic, sum of cubes formula, math problem explained, learn modular arithmetic, cube sum modulo 7", "---", "Meta Description:\nDiscover how to find the remainder when $1^3 + 2^3 + \dots + 10^3$ is divided by 7. Learn the cube sum formula, modular arithmetic steps, and why the result is $ \boxed{1} $. Ideal for math students and problem-solvers.", "---", "SQL Optimization Hint (for structured data):\nIndexing the cube sum formula by $n$ and precomputing mod values modulo 7 can improve performance in large-scale educational databases analyzing number patterns.", "---", "Author bio / SEO note:\nThis SEO-optimized article targets academic and student audiences seeking clear, accurate solutions to modular arithmetic problems. Correct, concise, and keyword-rich for search engines and human learning."]
