Question: What is the remainder when the sum $1^3 + 2^3 + 3^3 + \dots + 10^3$ is divided by $9$?
["SEO Optimized Article: Find the Remainder When $1^3 + 2^3 + \dots + 10^3$ is Divided by 9", "If you're curious about how modular arithmetic works with sums of cubes, you're in the right place. Today, we dive into a classic math problem: What is the remainder when the sum $1^3 + 2^3 + 3^3 + \dots + 10^3$ is divided by 9? This inquiry not only sharpens your understanding of number patterns and cubic sequences but also showcases the elegance of modular arithmetic.", "---", "### Introduction to the Sum of Cubes", "The expression $1^3 + 2^3 + 3^3 + \dots + n^3$ is a well-known summation in mathematics. A key identity states:", "[\n1^3 + 2^3 + 3^3 + \dots + n^3 = \left( \frac{n(n+1)}{2} \right)^2\n]", "This identity simplifies computation significantly and is the foundation for efficiently evaluating the cube sum. For $n = 10$, we begin by computing:", "[\n\left( \frac{10 \cdot 11}{2} \right)^2 = (55)^2 = 3025\n]", "So,\n[\n1^3 + 2^3 + \dots + 10^3 = 3025\n]", "---", "### Compute $3025 \mod 9$", "Now, the core question becomes: What is the remainder when 3025 is divided by 9?", "Rather than performing division, we use a powerful shortcut rooted in number theory:\nThe remainder of a number modulo 9 is equal to the sum of its digits modulo 9.", "This works because $10 \equiv 1 \pmod{9}$, so each digit’s place value contributes equally modulo 9.", "#### Step-by-step digit sum:", "$3 + 0 + 2 + 5 = 10$\nNow $1 + 0 = 1$", "So,\n[\n3025 \equiv 1 \pmod{9}\n]", "---", "### Verification via Direct Division", "For full clarity, dividing 3025 by 9:", "[\n3025 \div 9 = 336 \ ext{ remainder } 1\n]", "Since $9 \ imes 336 = 3024$,\n[\n3025 - 3024 = 1\n]", "Confirmed: remainder is indeed 1.", "---", "### Alternative Insight: Cubic Sums Modulo 9", "Beyond a one-off calculation, understanding cubic sums modulo 9 reveals a deeper pattern:", "All cubes modulo 9 repeat in cycles. Checking $k^3 \mod 9$ for $k = 0$ to $8$:", "- $0^3 \equiv 0$\n- $1^3 \equiv 1$\n- $2^3 = 8 \equiv 8$\n- $3^3 = 27 \equiv 0$\n- $4^3 = 64 \equiv 1$\n- $5^3 = 125 \equiv 8$\n- $6^3 = 216 \equiv 0$\n- $7^3 = 343 \equiv 1$\n- $8^3 = 512 \equiv 8$", "So cube residues mod 9: $0, 1, 8, 0, 1, 8, 0, 1, 8$", "Now sum one cycle:\n$0 + 1 + 8 + 0 + 1 + 8 + 0 + 1 + 8 = 27 \equiv 0 \pmod{9}$", "From $k = 1$ to $9$, the residues repeat every 3 terms with pattern $1, 8, 0$ summing to 9. For $k = 1$ to $9$, sum mod 9 is 0. But we go to $k = 10$, adding $10^3 \equiv 1^3 \equiv 1 \pmod{9}$. So total sum mod 9 = $0 + 1 = 1$.", "---", "### Conclusion", "Whether using the algebraic identity, digit sum trick, or modular patterns, the remainder when $1^3 + 2^3 + \dots + 10^3$ is divided by 9 is:", "[\n\boxed{1}\n]", "This problem exemplifies how modular arithmetic simplifies seemingly complex sums and highlights recurring number patterns—an essential skill in mathematics education and competitive problem-solving.", "---", "Tagline for SEO:\n Learn how to compute $1^3 + 2^3 + \dots + 10^3 \mod 9$, find the remainder, use digit sums, and explore modular patterns—examples of smart math tricks for students and enthusiasts.", "Keywords: remainder when sum of cubes divided by 9, $1^3 + 2^3 + \dots + 10^3 \mod 9$, sum of cubes formula, modular arithmetic examples, digit sum trick, math problem solving, cubic sum modulo 9", "---", "Optimized for search engines with clear headings, structured content, and relevant keywords to help learners and educators quickly grasp and apply this classic modular arithmetic problem."]
