Problem 1 - Prealgebra

Question

Find the GCF of each set of numbers. $$6,8$$

Step-by-Step Solution

Answer

The GCF of 6 and 8 is 2.

Step 1: Understand the Concept

The Greatest Common Factor (GCF) of two numbers is the largest number that divides both of them without leaving a remainder. The GCF is useful in simplifying fractions or finding common denominators.

Step 2: List the Factors

List all the factors of each number. For 6, the factors are 1, 2, 3, 6. For 8, the factors are 1, 2, 4, 8.

Step 3: Identify Common Factors

Compare the lists of factors to find common factors. The common factors of 6 and 8 are 1 and 2.

Step 4: Determine the GCF

Select the largest common factor from the list of common factors. The largest number in common between the factors of 6 and 8 is 2.

Key Concepts

Factors

When we talk about numbers in mathematics, factors are numbers that you can multiply together to get another number. Factors play a crucial role in various mathematical operations, including finding the Greatest Common Factor (GCF). To break it down:

• A factor of a number is an exact divisor of that number — meaning when the number is divided by its factor, it leaves no remainder.
• For example, consider the number 8. Its factors are 1, 2, 4, and 8 since all these numbers can divide 8 without any remainder.
• It's also important to know that every number has at least two factors: 1 and the number itself.

Understanding factors is crucial because they form the basis for operations such as simplifying fractions and finding the GCF, which involves comparing lists of factors from different numbers to find the largest one they share.

Prime Factorization

Prime factorization involves expressing a number as the product of its prime factors. Prime factors are the building blocks of a number and are essential in finding the Greatest Common Factor (GCF).

• Prime numbers are numbers greater than 1 with no divisors other than 1 and themselves. Examples include 2, 3, 5, and 7.
• Prime factorization is important because it breaks down numbers into their smallest divisible units. For instance, the number 6 can be expressed as the product of primes: 2 and 3, thus 6 = 2 \(\times\) 3.

To find the GCF using prime factorization, you would:

• Factorize each number into its prime factors.
• Identify the common prime factors and choose the ones with the smallest power shared by all numbers.
• Multiply these common prime factors to determine the GCF.

This method is particularly helpful with larger numbers where listing all factors can become cumbersome.

Fractions

Fractions represent a part of a whole and are expressed as \(\frac{a}{b}\), where \(a\) is the numerator and \(b\) is the denominator. Simplifying fractions often involves using the Greatest Common Factor (GCF).

• To simplify a fraction, divide both the numerator and the denominator by their greatest common factor.
• For example, to simplify the fraction \(\frac{6}{8}\), find the GCF of 6 and 8, which is 2. Then divide both the numerator and the denominator by 2, leaving \(\frac{3}{4}\).
• This simplification helps in reducing fractions to their simplest form, making calculations and comparisons easier.

Working with fractions and understanding their simplification is integral to many areas of math, including arithmetic operations, algebraic equations, and problem-solving in real-life scenarios.