Therefore, the greatest common divisor of 420 and 630 is \(\boxed{210}\).### Question 1

["Discover Why the Greatest Common Divisor of 420 and 630 Is 210 – A Detailed Explanation", "When tackling math problems involving large numbers, understanding the greatest common divisor (GCD) is essential. One frequently encountered question is: What is the greatest common divisor of 420 and 630? The correct answer is (\boxed{210})—but why exactly?", "### What Is the Greatest Common Divisor (GCD)?", "The GCD of two integers is the largest positive number that divides both without leaving a remainder. It’s widely used in simplifying fractions, solving divisibility problems, and optimizing real-world scenarios like dividing resources or scheduling repeating events.", "### Step-by-Step: Finding GCD(420, 630)", "To compute (\ ext{GCD}(420, 630)), we use two powerful methods: the prime factorization technique and Euclid’s algorithm. Both confirm the same result—and a clean, elegant solution.", "#### Method 1: Prime Factorization", "Factorize both numbers into their prime components:", "- (420 = 2^2 \ imes 3 \ imes 5 \ imes 7)\n- (630 = 2 \ imes 3^2 \ imes 5 \ imes 7)", "Identify the common prime factors with their lowest powers:\n- (2^1), (3^1), (5^1), (7^1)", "Multiply these together:\n(2 \ imes 3 \ imes 5 \ imes 7 = 210)", "Thus, the GCD is 210.", "#### Method 2: Euclid’s Algorithm", "This efficient method uses repeated division:", "1. Divide the larger number (630) by the smaller (420):\n (630 \div 420 = 1) with remainder (210).", "2. Replace the larger number with the smaller, and the smaller with the remainder:\n Now compute (\ ext{GCD}(420, 210)).", "3. Repeat:\n (420 \div 210 = 2) with remainder (0).", "When the remainder hits zero, the last non-zero remainder is the GCD.", "➡️ Therefore, (\ ext{GCD}(420, 630) = 210).", "### Why 210 Matters", "- Simplification: (\frac{420}{210} = 2) and (\frac{630}{210} = 3), breaking 420 and 630 into smaller, manageable parts.\n- LCM Calculation: The least common multiple of 420 and 630 is (\frac{420 \ imes 630}{\ ext{GCD}} = 12600), useful for finding shared cycles or combined periods.\n- Real-World Applications: Whether splitting 420 items into 210 equal groups or scheduling events recurring every 420 and 630 minutes, 210 ensures fairness and efficiency.", "### Summary", "Both mathematical approaches confirm that the greatest common divisor of 420 and 630 is unequivocally (\boxed{210}). This number isn’t just a result—it’s a key to unlocking deeper understanding in number theory and practical problem-solving.", "Mastering the GCD of 420 and 630 prepares you for more advanced math challenges and highlights how simple numbers govern complex systems in science, engineering, and everyday life. Start practicing with other pairs—you’ll consistently find 210 or another elegant divisor helping you succeed."]