What is the greatest common factor of 78 and 91?
["# What Is the Greatest Common Factor of 78 and 91?", "Understanding the greatest common factor (GCF) of two numbers is fundamental in math, especially in learning about divisibility, fractions, and simplifying ratios. One commonly studied pair of numbers is 78 and 91. But what exactly is the GCF, and what is it for 78 and 91? Let’s explore it clearly and simply.", "## What Is the Greatest Common Factor (GCF)?", "The greatest common factor (GCF), also known as the greatest common divisor (GCD), is the largest positive integer that divides two or more numbers exactly without leaving a remainder. In other words, it’s the biggest number that can evenly divide both numbers.", "For example, while 78 and 91 may initially seem unrelated, identifying their GCF helps reveal underlying relationships between them and simplifies problems involving fractions, ratios, and division.", "## Step-by-Step: Finding the GCF of 78 and 91", "To find the GCF of 78 and 91, we use the prime factorization method, which breaks each number into its prime components.", "### Step 1: Prime Factorization\nBreak down 78 and 91 into prime factors:", "- 78:\n 78 ÷ 2 = 39\n 39 ÷ 3 = 13\n 13 is prime\n So, $ 78 = 2 \ imes 3 \ imes 13 $", "- 91:\n 91 ÷ 7 = 13\n 13 is prime\n So, $ 91 = 7 \ imes 13 $", "### Step 2: Identify Common Prime Factors\nNow compare the prime factorizations:", "- 78 = 2 × 3 × 13\n- 91 = 7 × 13", "The only prime number both share is 13.", "### Step 3: Multiply the Common Factors\nSince 13 appears in both factorizations, the GCF is:", "$$\n\ ext{GCF}(78, 91) = 13\n$$", "## Why Is the GCF Important?", "Knowing the GCF of 78 and 91 helps in various mathematical applications:", "- Simplifying fractions: Though 78 and 91 don’t simplify a common fraction directly, understanding GCF improves fraction comparison and simplification skills.\n- Working with ratios: Ratios like 78:91 can be simplified if divided by their GCF, revealing equivalent ratios.\n- GCD applications: From dividing resources evenly to solving LCM problems, GCF is a foundational concept in number theory.", "## Summary", "- The greatest common factor of 78 and 91 is 13.\n- 13 is the largest number that divides both 78 and 91 evenly.\n- Finding the GCF involves prime factorization and identifying shared prime factors.\n- This knowledge supports stronger problem-solving in fractions, ratios, and algebra.", "---", "From math classrooms to real-life calculations, understanding the GCF empowers accurate and efficient problem-solving. Whether you’re tackling homework or everyday math challenges, recognizing that 13 is the greatest common factor of 78 and 91 comes in handy every time!"]
