2Question: What is the remainder when $2025 + 2027 + 2029 + 2031$ is divided by 12?
["SEO-Optimized Article: What is the Remainder When $2025 + 2027 + 2029 + 2031$ is Divided by 12?", "Calculating the remainder when $2025 + 2027 + 2029 + 2031$ is divided by 12 may seem daunting at first, but with a few strategic steps, it becomes simple and insightful. Whether you're a student learning modular arithmetic or someone curious about number patterns, understanding how to find remainders efficiently is essential.", "### Step-by-Step Breakdown of the Sum", "First, let’s examine the numbers:\n$2025$, $2027$, $2029$, and $2031$ are four consecutive odd numbers. They form an arithmetic sequence with a common difference of 2.", "Compute their sum:\n[\n2025 + 2027 + 2029 + 2031\n]", "Add them step-by-step:\n[\n(2025 + 2031) + (2027 + 2029) = 4056 + 4056 = 8112\n]", "So, the total sum is $8112$. Now, we want to find:\n[\n8112 \mod 12\n]", "### Efficient Way Using Modular Arithmetic", "Instead of dividing 8112 by 12 directly, use properties of modular arithmetic to simplify:", "Note that:\n[\n2025 \mod 12,\ 2027 \mod 12,\ 2029 \mod 12,\ 2031 \mod 12\n]", "Instead of computing each separately, observe that all four numbers are congruent modulo 12 with respect to their offset from a multiple of 12.", "But a faster method is to use the fact that for any number $n$, if we know $n \mod 12$, we can compute it by dividing stepwise.", "Alternatively, divide the entire sum by 12:", "[\n8112 \div 12 = 676 \quad \ ext{(exactly, since } 12 \ imes 676 = 8112\ ext{)}\n]", "Since 8112 is exactly divisible by 12:\n[\n8112 \mod 12 = 0\n]", "### Final Answer Explained", "Thus, the remainder when $2025 + 2027 + 2029 + 2031$ is divided by 12 is 0.", "This result confirms that the sum of these four consecutive odd numbers is perfectly divisible by 12.", "### Why This Matters", "Understanding remainders in modular arithmetic helps in cryptography, computer science, calendar calculations, and more. The sum being divisible by 12 reveals a subtle symmetry — four odd numbers spaced by 2 add up to a multiple of 12.", "TL;DR:\nThe remainder when $2025 + 2027 + 2029 + 2031$ is divided by 12 is 0 — it divides evenly with no remainder.", "---", "Keywords: remainder when 2025+2027+2029+2031 divided by 12, solve 2025+2027+2029+2031 mod 12, modular arithmetic example, sum of consecutive odd numbers mod 12, math problem solution", "Meta Description:\nFind the remainder of $2025 + 2027 + 2029 + 2031$ when divided by 12 using step-by-step modular arithmetic — the answer is 0. Learn how to simplify such sums quickly with step-by-step explanation."]
