Solution: The sum of cubes formula is $\left(\frac{n(n+1)}{2}\right)^2$. For $n = 12$, the sum is $\left(\frac{12 \cdot 13}{2}\right)^2 = (78)^2 = 6084$. Dividing by 13: $6084 \div 13 = 468$ with remainder 0, since $13 \cdot 468 = 6084$.

["Mastering the Sum of Cubes Formula: Proving $\left(\frac{n(n+1)}{2}\right)^2 = 6084$ for $n = 12$", "The sum of the first $n$ cubes has fascinated mathematicians for centuries, and one of the most elegant and powerful formulas in mathematics is:", "$$\n\sum_{k=1}^{n} k^3 = \left( \frac{n(n+1)}{2} \right)^2\n$$", "This formula reveals that the sum of the cubes of the first $n$ positive integers is itself a perfect square — the square of the $n$th triangular number.", "### What Is the Sum of the First 12 Cubes?", "Let’s take a concrete example: compute the sum $\sum_{k=1}^{12} k^3$. Applying the sum of cubes formula:", "$$\n\left( \frac{n(n+1)}{2} \right)^2 = \left( \frac{12 \cdot 13}{2} \right)^2 = (78)^2 = 6084\n$$", "This powerful shortcut replaces the labor of adding 12 terms individually — a method that would take time and effort — with a single calculation.", "### Verifying the Result: Divide by 13", "To strengthen understanding, consider dividing the sum (6084) by 13:", "$$\n6084 \div 13 = 468 \quad \ ext{with no remainder}\n$$", "This confirms that 6084 is exactly divisible by 13, yielding the integer 468. This divisibility check reinforces how deeply connected these numbers are through arithmetic patterns.", "Indeed, $13 \ imes 468 = 6084$, proving the consistency and accuracy of the formula.", "### Why This Formula Matters", "Beyond computational convenience, this identity connects algebra, number theory, and geometry. Its geometric interpretation — expressed by square numbers matching cumulative cubic sums — showcases the harmony of mathematical concepts. It also serves as a foundational tool in proofs and advanced problem-solving.", "### Conclusion", "The sum of the cubes of the first $n$ integers, elegantly captured by $\left( \frac{n(n+1)}{2} \right)^2$, transforms seemingly complex summation into a simple square. For $n = 12$, this formula delivers $6084$, a number divisible cleanly by 13, validating both calculation and mathematical structure. Whether learning, teaching, or solving advanced problems, mastering this formula unlocks deeper insights into patterns of numbers.", "---", "Key SEO terms: sum of cubes formula, mathematical formula proof, $\sum k^3$ identity, triangular number square, divisibility by 13, nth cube sum formula, perfect square identity, algebraic summation formula, math education, number theory examples."]