\boxed{1}Question: What is the greatest common factor of 48 and 60?

["# The Greatest Common Factor of 48 and 60: A Clear Breakdown", "When learning about factors in mathematics, one essential concept is the greatest common factor (GCF), also known as the greatest common divisor (GCD). Understanding the GCF helps simplify fractions, solve equations, and work more effectively with numbers. But a common question arises: What is the greatest common factor of 48 and 60? This helpful guide explains how to find the GCF using multiple methods, making it easy to determine that the answer is 12.", "## What is the Greatest Common Factor?", "The greatest common factor of two or more numbers is the largest positive integer that divides each number evenly without leaving a remainder. For example, the GCF of 48 and 60 is the biggest number that divides both without a remainder.", "### Why Finding the GCF Matters", "- Simplifies fractions\n- Reduces complex calculations\n- Enhances number sense and algebraic thinking", "---", "## Step-by-Step Methods to Find the GCF of 48 and 60", "There are several straightforward ways to determine the GCF of 48 and 60:", "### Method 1: Listing Factors", "A simple way is to list all positive factors for each number and identify the largest common one.", "Factors of 48:\n1, 2, 3, 4, 6, 8, 12, 16, 24, 48", "Factors of 60:\n1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60", "Common factors: 1, 2, 3, 4, 6, 12\nThe largest is 12.", "---", "### Method 2: Prime Factorization", "Breaking each number into its prime factors makes comparing factors clear and efficient.", "Prime factorization of 48:\n48 = 2 × 2 × 2 × 2 × 3 = (2^4 × 3^1)", "Prime factorization of 60:\n60 = 2 × 2 × 3 × 5 = (2^2 × 3^1 × 5^1)", "To find the GCF, take the lowest power of each common prime factor:", "- Common primes: 2 and 3\n- Minimum powers: (2^2) and (3^1)\n- Multiply: (2^2 × 3 = 4 × 3 = 12)", "GCF = 12", "---", "### Method 3: Using the Euclidean Algorithm", "For larger numbers, the Euclidean algorithm efficiently finds the GCF through repeated division.", "Divide 60 by 48:\n60 ÷ 48 = 1 remainder 12", "Now divide 48 by 12:\n48 ÷ 12 = 4 remainder 0", "When the remainder is 0, the last non-zero remainder is the GCF.\nThus, GCF(48, 60) = 12", "---", "## Why the GCF of 48 and 60 is 12", "Once the GCF is found to be 12, we can verify:", "- 48 ÷ 12 = 4 (whole number)\n- 60 ÷ 12 = 5 (whole number)", "12 divides both 48 and 60 perfectly, and no larger number does, confirming it is the greatest common factor.", "---", "## Practical Applications of the GCF of 48 and 60", "Knowing the GCF is useful in everyday math:", "- Simplifying fractions: For instance, (\frac{48}{60}) simplifies to (\frac{4}{5}) by dividing numerator and denominator by 12.\n- Finding least common multiples (LCM): GCF helps calculate LCM using the formula: LCM(a, b) = (\frac{a × b}{GCF(a, b)}).\n- Teaching children division and ratios with ease.", "---", "## Summary", "The greatest common factor of 48 and 60 is 12. Whether you list factors, factor into primes, or apply the Euclidean algorithm, each method confirms this result. Understanding the GCF strengthens your mathematical foundation and supports a range of real-world calculations.", "If you often encounter mixed numbers or complex fractions, mastering GCF calculations like this builds confidence and skill for more advanced math topics.", "Key takeaway: The GCF of 48 and 60 is 12 — a fundamental number that simplifies fractions and supports clearer mathematical reasoning."]