Problem 14 - Prealgebra

Question

Find the GCF of each set of numbers or monomials. $$12,8$$

Step-by-Step Solution

Answer

The GCF of 12 and 8 is 4.

Step 1: List the Factors of Each Number

First, write down all the factors of each number. For 12, the factors are 1, 2, 3, 4, 6, and 12. For 8, the factors are 1, 2, 4, and 8.

Step 2: Identify the Common Factors

Look at the list of factors for both 12 and 8. The factors that both numbers share are 1, 2, and 4.

Step 3: Find the Greatest Common Factor

Out of the common factors identified in Step 2, the greatest one is 4. Therefore, the greatest common factor (GCF) of 12 and 8 is 4.

Key Concepts

Understanding Factors

Factors are numbers you can multiply together to get another number. They are like building blocks for a number. For example, with the number 12, if you multiply 2 by 6, you get 12. This means both 2 and 6 are factors of 12.
When searching for factors, you're looking for all whole numbers that divide evenly into the original number. Here are some practical tips:

• Start with the number 1, because 1 is a factor of every number.
• Move upwards, testing each whole number until you reach the original number.
• Write down each whole number that evenly divides your original number.

By listing these factors, you get a clearer picture of what numbers form your original number when multiplied together.

Unveiling Common Factors

Common factors are the numbers that appear in the factor lists of two or more numbers. To find them, first, gather the factors of each individual number. Next, cross-reference these lists to find numbers that both share.

Common Factor Example

Consider finding the common factors of 12 and 8. First, identify all the factors:

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 8: 1, 2, 4, 8

Common factors exist in both lists. Look carefully and you’ll see that 1, 2, and 4 stand out as common factors. These numbers divide both 12 and 8 without leaving any remainder. Recognizing common factors helps simplify and solve various mathematical problems.

Performing Numerical Operations with GCF

Numerical operations involving the greatest common factor (GCF) help simplify or solve number scenarios more efficiently. The GCF is the largest number that can evenly divide two or more numbers, helping in reducing fractions or finding solutions in equations.

Applying GCF in Real Life

When faced with the task to simplify fractions, the GCF is essential. Consider the fraction 12/8. Knowing the GCF of 12 and 8 is 4 helps to reduce this fraction. Divide both the numerator and the denominator by 4:

• Numerator: 12 ÷ 4 = 3
• Denominator: 8 ÷ 4 = 2

Thus, the fraction 12/8 reduces to 3/2. The GCF not only aids in fraction reduction but is useful in solving many mathematical problems, making calculations neater and more manageable.