Question: What is the remainder when $1^2 + 2^2 + \cdots + 15^2$ is divided by 13?
["Title: Quick Way to Compute Remainder: Sum of Squares $1^2 + 2^2 + \cdots + 15^2$ Modulo 13", "Understanding how to compute the remainder of a sum of squares modulo a number efficiently is a valuable skill in number theory and competitive math. One common question students encounter is: What is the remainder when $1^2 + 2^2 + \cdots + 15^2$ is divided by 13? This article explains how to solve this problem using mathematical formulas and modular arithmetic for a quick and accurate result.", "---", "### Understanding the Formula for Sum of Squares", "The sum of the squares of the first $n$ natural numbers is given by the well-known formula:\n[\nS = 1^2 + 2^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}\n]", "For $n = 15$, plugging in the number:\n[\nS = \frac{15 \cdot 16 \cdot 31}{6}\n]", "Computing this directly can be messy, especially under a modulus. Instead, we compute the remainder modulo 13 using modular arithmetic properties to simplify calculations.", "---", "### Step 1: Reduce Each Component Modulo 13", "Instead of calculating the full sum first, we reduce each factor modulo 13:", "- $15 \mod 13 = 2$\n- $16 \mod 13 = 3$\n- $31 \mod 13 = 5$ (since $31 - 2\ imes13 = 31 - 26 = 5$)", "So the numerator becomes:\n[\nn(n+1)(2n+1) \mod 13 = 2 \cdot 3 \cdot 5 = 30 \equiv 4 \pmod{13}\n]", "---", "### Step 2: Compute Denominator Modulo 13", "The denominator is 6. We need the modular inverse of 6 modulo 13.", "Find $6^{-1} \mod 13$: find a number $x$ such that $6x \equiv 1 \pmod{13}$.\nTesting values:\n$6 \cdot 11 = 66 \equiv 1 \pmod{13}$ (since $66 \div 13 = 5$ remainder 1)\nThus, $6^{-1} \equiv 11 \pmod{13}$", "---", "### Step 3: Combine Results Modulo 13", "Now compute:\n[\nS \mod 13 = \left( \frac{2 \cdot 3 \cdot 5}{6} \right) \mod 13 = (4 \cdot 11) \mod 13 = 44 \mod 13\n]", "Finally, $44 \div 13 = 3$ remainder $5$, so:\n[\n44 \mod 13 = 5\n]", "---", "### Conclusion", "The remainder when $1^2 + 2^2 + \cdots + 15^2$ is divided by 13 is $\boxed{5}$.", "This approach avoids large intermediate values, uses efficient modular arithmetic, and leverages the formula for the sum of squares — a straightforward method for solving this type of problem quickly and confidently.", "---", "Keywords:\nremainder of $1^2 + 2^2 + \cdots + 15^2$ divided by 13, sum of squares modulo 13, modular arithmetic, formula for sum of squares, quick calculation method, number theory tips", "Meta Description:\nLearn how to compute the remainder when $1^2 + 2^2 + \cdots + 15^2$ is divided by 13 using the sum of squares formula and modular arithmetic for fast, accurate results."]
