Question: What is the remainder when the sum $2^3 + 4^3 + 6^3 + \dots + 10^3$ is divided by 9?
["SEO-Optimized Article: Finding the Remainder of the Sum $2^3 + 4^3 + 6^3 + \dots + 10^3$ When Divided by 9", "Understanding how to compute remainders in modular arithmetic is essential in number theory and competitive math. Today, we explore a specific cuestión: What is the remainder when the sum $2^3 + 4^3 + 6^3 + \dots + 10^3$ is divided by 9?", "### Breaking Down the Problem", "The sum involves the cubes of even numbers from 2 to 10:\n[\n2^3 + 4^3 + 6^3 + 8^3 + 10^3\n]", "Note these numbers form an arithmetic sequence: $2, 4, 6, 8, 10$, all even. This sequence has 5 terms with a common difference of 2. We can rewrite each term as $ (2k)^3 $ for $ k = 1 $ to $ 5 $:\n[\n(2 \cdot 1)^3 + (2 \cdot 2)^3 + (2 \cdot 3)^3 + (2 \cdot 4)^3 + (2 \cdot 5)^3 = 8(1^3 + 2^3 + 3^3 + 4^3 + 5^3)\n]", "Now we compute the sum inside the parentheses:\n[\n1^3 + 2^3 + 3^3 + 4^3 + 5^3 = 1 + 8 + 27 + 64 + 125 = 225\n]", "So the full sum becomes:\n[\n8 \ imes 225 = 1800\n]", "### Finding the Remainder When Divided by 9", "We now compute:\n[\n1800 \mod 9\n]", "A powerful shortcut: The remainder of a number modulo 9 equals the remainder of the sum of its digits modulo 9.\nSum the digits of 1800:\n[\n1 + 8 + 0 + 0 = 9\n]", "Then:\n[\n9 \mod 9 = 0\n]", "Thus,\n[\n1800 \equiv 0 \pmod{9}\n]", "### Conclusion", "The remainder when the sum $2^3 + 4^3 + 6^3 + 8^3 + 10^3$ is divided by 9 is 0.", "This problem highlights how modular arithmetic can simplify calculations involving powers and sequences. Knowing these patterns not only solves specific questions efficiently but also strengthens foundational skills in number theory and divisibility rules.", "Keyword-rich summary:\n- modular arithmetic\n- remainders mod 9\n- sum of cubes\n- arithmetic sequence cubes\n- number theory applications\n- divisibility by 9", "Optimize your next math challenge by recognizing patterns and applying smart shortcuts—like using digit sums—to unlock efficient solutions!"]
