What is the remainder when the sum \( 1^3 + 2^3 + 3^3 + \dots + 10^3 \) is divided by 11?
["What is the Remainder When the Sum ( 1^3 + 2^3 + 3^3 + \dots + 10^3 ) is Divided by 11?", "Calculating the sum of cubes from ( 1^3 ) to ( 10^3 ) and finding the remainder modulo 11 is a classic math problem with elegant applications in number theory and modular arithmetic. In this article, we explore how to evaluate this sum efficiently and determine its remainder when divided by 11.", "---", "### Understanding the Sum of Cubes Formula", "The sum of the cubes of the first ( n ) natural numbers is given by the formula:\n[\n1^3 + 2^3 + 3^3 + \dots + n^3 = \left( \frac{n(n+1)}{2} \right)^2\n]", "For ( n = 10 ), the sum becomes:\n[\n\left( \frac{10 \cdot 11}{2} \right)^2 = (55)^2 = 3025\n]", "So,\n[\n1^3 + 2^3 + \dots + 10^3 = 3025\n]", "---", "### Finding the Remainder When 3025 Is Divided by 11", "To find ( 3025 \mod 11 ), we compute:\n[\n3025 \div 11\n]", "We can simplify this using modular arithmetic. Since 11 is prime, properties of modular arithmetic help.", "#### Step-by-step calculation:", "Divide 3025 by 11:\n[\n11 \ imes 275 = 3025\n]", "Thus,\n[\n3025 \equiv 0 \pmod{11}\n]", "So the remainder is 0.", "---", "### Why Is the Remainder Zero?", "An insightful observation connects this result to congruences and symmetry modulo 11.", "The sum ( 1^3 + 2^3 + \dots + 10^3 ) involves cubes of all nonzero residues modulo 11 (since 0³ is 0, but 0 is not included). Because 11 is prime, the set ( {1, 2, \dots, 10} ) forms the complete residue system modulo 11 excluding 0. The sum of all nonzero cubes modulo a prime can sometimes be zero due to symmetry or algebraic identities.", "Indeed, a known identity states:\nFor prime ( p \equiv 3 \pmod{4} ) or ( p \equiv 1 \pmod{4} ), the sum ( \sum_{k=1}^{p-1} k^3 \equiv 0 \pmod{p} ) — but more directly, since ( \left( \frac{n(n+1)}{2} \right)^2 ) is a perfect square and divisibility by 11 depends on topology in mod 11, direct computation confirms ( 55^2 \equiv 0 \pmod{11} ), since ( 55 ) is divisible by 11.", "---", "### Practical Applications of This Computation", "This problem illustrates how number theory underpins efficient evaluations in computer science, cryptography, and algorithm design — for example, fast modular exponentiation and verifying divisibility properties.", "---", "### Final Answer", "The remainder when ( 1^3 + 2^3 + 3^3 + \dots + 10^3 ) is divided by 11 is:\n[\n\boxed{0}\n]", "---", "### Key takeaways:\n- Use the cube sum formula to simplify large sums.\n- Use modular arithmetic to compute remainders efficiently.\n- Recognition that symmetric residue systems modulo primes often yield zero sums in cubic forms.\n- This classic example bridges discreet math and computational thinking."]
