2Question: What is the remainder when the sum of the first 100 even numbers is divided by 7?
["Understanding the Remainder: First 100 Even Numbers Modulo 7", "When solving mathematical puzzles, one intriguing question often arises: What is the remainder when the sum of the first 100 even numbers is divided by 7? This question combines basic arithmetic progressions with modular arithmetic — a practical and enlightening topic for students, teachers, and math enthusiasts alike.", "### What Are the First 100 Even Numbers?", "The sequence of even numbers starts from 2 and continues as 2, 4, 6, 8, ..., up to the 100th even number. This sequence is an arithmetic progression where:", "- First term ((a_1)) = 2\n- Common difference ((d)) = 2\n- Number of terms ((n)) = 100", "### Calculating the Sum of the First 100 Even Numbers", "The sum (S_n) of the first (n) terms of an arithmetic sequence is given by:\n[\nS_n = \frac{n}{2} \cdot (a_1 + a_n)\n]", "To apply this formula, we need the 100th even number:\n[\na_{100} = 2 + (100 - 1) \cdot 2 = 2 + 198 = 200\n]", "Now substitute into the sum formula:\n[\nS_{100} = \frac{100}{2} \cdot (2 + 200) = 50 \cdot 202 = 10100\n]", "### Finding the Remainder When Divided by 7", "Next, determine:\n[\n10100 \mod 7\n]", "To simplify this calculation, divide 10100 by 7 and find the remainder:", "Use long division or modular shortcuts. First, observe:\n[\n10100 \div 7 = 1442 \ ext{ remainder } r\n]", "We can compute directly:\n[\n7 \ imes 1442 = 10094\n]\n[\n10100 - 10094 = 6\n]", "Therefore,\n[\n10100 \mod 7 = 6\n]", "### Conclusion: The Remainder Is 6", "Final Answer: The remainder when the sum of the first 100 even numbers is divided by 7 is 6.", "---", "### Why This Problem Matters", "This question elegantly blends pattern recognition, arithmetic series, and modular arithmetic — foundational skills in number theory and real-world applications like coding, cryptography, and calendar calculations. Understanding how large sums behave modulo small numbers enhances problem-solving precision and deepens mathematical intuition.", "For learners and educators, exploring this problem offers both a concrete calculation exercise and a gateway to broader number theory concepts.", "---", "Keywords: remainder when sum of first 100 even numbers divided by 7, even numbers sum modulo 7, mathematical puzzles, arithmetic series sum mod, modular arithmetic example, 10100 remainder 7, math problem solution.", "---", "Learn more about modular arithmetic and number patterns to solve interesting math challenges effortlessly."]
