Question: What is the greatest common factor of 68 and 102? - Decision Point
What is the Greatest Common Factor of 68 and 102?
What is the Greatest Common Factor of 68 and 102?
Understanding mathematics often starts with simple—but essential—concepts, one of which is the Greatest Common Factor, or GCF. If you’ve ever asked, “What is the greatest common factor of 68 and 102?, you’re not alone. Finding the GCF helps solve everything from basic algebra to real-world division problems. In this article, we’ll break down how to find the GCF of 68 and 102, why it matters, and how to do it quickly and accurately.
What Is the Greatest Common Factor?
Understanding the Context
The Greatest Common Factor (GCF), also known as the Greatest Common Divisor (GCD), is the largest positive number that divides two or more integers without leaving a remainder. It’s a fundamental tool in number theory and is widely used in simplifying fractions, factoring numbers, and solving practical problems.
Why Does GCD Matter?
Knowing the GCF helps you:
- Simplify fractions to their lowest terms
- Determine whether two numbers are relatively prime
- Solve problems involving ratios and proportions
- Facilitate easier multiplication and division of large numbers
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How to Find the GCF of 68 and 102
Let’s walk through the step-by-step process of finding the GCF of 68 and 102 using the most reliable method: prime factorization.
Step 1: Prime Factorization of Each Number
Break each number into its prime factors:
- 68
68 ÷ 2 = 34
34 ÷ 2 = 17
17 is prime
⇒ 68 = 2 × 2 × 17 = 2² × 17
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- 102
102 ÷ 2 = 51
51 ÷ 3 = 17
17 is prime
⇒ 102 = 2 × 3 × 17
Step 2: Identify Common Prime Factors
Compare the prime factorizations:
- 68 = 2² × 17
- 102 = 2 × 3 × 17
The common prime factors are 2 and 17.
Step 3: Multiply the Lowest Powers of Common Factors
- The common factor 2 appears to the 1st power in 102 (and 2² in 68), so we take 2¹ = 2
- The factor 17 appears to the 1st power in both, so we take it as 17¹ = 17
Multiply them:
GCF = 2 × 17 = 34
Verification: Does 34 Divide Both Numbers?
- 68 ÷ 34 = 2 (whole number)
- 102 ÷ 34 = 3 (whole number)
