Grade 4
  1. Numbers and Place Value  1.1. Understanding Place Value  1.2. Reading and Writing Large Numbers  1.3. Comparing and Ordering Numbers  1.4. Rounding Numbers  1.5. Understanding Odd and Even Numbers  1.6. Number Patterns and Sequences
  2. Addition and Subtraction  2.1. Addition of Large Numbers  2.2. Subtraction of Large Numbers  2.3. Estimation in Addition and Subtraction  2.4. Word Problems on Addition and Subtraction
  3. Multiplication and Division  3.1. Understanding Multiplication  3.2. Multiplying Multi-Digit Numbers  3.3. Understanding Division  3.4. Dividing Multi-Digit Numbers  3.5. Multiplication and Division Facts  3.6. Word Problems on Multiplication and Division
  4. Factors and Multiples  4.1. Understanding Factors  4.2. Finding Common Factors  4.3. Understanding Multiples  4.4. Finding Common Multiples  4.5. Prime and Composite Numbers  4.6. Prime Factorization  4.7. Greatest Common Factor  4.8. Least Common Multiple
  5. Fractions  5.1. Understanding Fractions  5.2. Equivalent Fractions  5.3. Simplifying Fractions  5.4. Comparing and Ordering Fractions  5.5. Addition of Fractions  5.6. Subtraction of Fractions  5.7. Mixed Numbers and Improper Fractions  5.8. Converting Mixed Numbers to Improper Fractions  5.9. Word Problems on Fractions
  6. Decimals  6.1. Understanding Decimals  6.2. Reading and Writing Decimals  6.3. Comparing and Ordering Decimals  6.4. Rounding Decimals  6.5. Addition of Decimals  6.6. Subtraction of Decimals  6.7. Converting Decimals to Fractions  6.8. Converting Fractions to Decimals  6.9. Word Problems on Decimals
  7. Measurement  7.1. Length  7.1.1. Units of Length  7.1.2. Converting Units of Length  7.1.3. Perimeter  7.1.4. Area  7.1.5. Word Problems on Length  7.2. Weight  7.2.1. Units of Weight  7.2.2. Converting Units of Weight  7.2.3. Word Problems on Weight  7.3. Volume and Capacity  7.3.1. Units of Volume  7.3.2. Converting Units of Volume  7.3.3. Estimating Volume and Capacity  7.3.4. Word Problems on Volume and Capacity  7.4. Time  7.4.1. Reading Time  7.4.2. Units of Time  7.4.3. Converting Units of Time  7.4.4. Calculating Elapsed Time  7.4.5. Word Problems on Time
  8. Geometry  8.1. Properties of Shapes  8.1.1. Types of Angles  8.1.2. Types of Triangles  8.1.4. Symmetry  8.1.5. Lines of Symmetry  8.2. Perimeter and Area  8.2.1. Perimeter of Rectangles  8.2.2. Perimeter of Other Shapes  8.2.3. Area of Rectangles  8.2.4. Area of Other Shapes  8.2.5. Area of Irregular Shapes  8.3. Coordinate Geometry  8.3.1. Understanding Coordinates  8.3.2. Plotting Points on a Grid  8.3.3. Identifying Shapes on a Grid
  9. Data and Graphs  9.1. Types of Data  9.2. Organizing Data  9.3. Reading Bar Graphs  9.4. Drawing Bar Graphs  9.5. Line Plots  9.7. Reading and Interpreting Line Graphs  9.8. Word Problems on Graphs
  10. Money  10.1. Understanding Money  10.2. Addition and Subtraction with Money  10.3. Making Change  10.4. Multiplying and Dividing Money  10.5. Word Problems on Money

Grade 4 → Factors and Multiples ↓


Greatest Common Factor


Welcome to the world of numbers! Today we are going to talk about something called the greatest common factor, or GCF for short. The greatest common factor is the largest number that can divide two or more numbers without leaving a remainder. It's a concept you encounter early on in math, and it's essential to understanding how numbers work together.

What are the factors?

Before we get into the greatest common factor, it is important to understand what factors are. A factor is a number that divides another number exactly, leaving no remainder. For example, consider the number 12. Do this. The factors of 12 are all the numbers that can divide 12 evenly. These numbers are 1, 2, 3, 4, 6, and 12. This means that:

  • 12 ÷ 1 = 12
  • 12 ÷ 2 = 6
  • 12 ÷ 3 = 4
  • 12 ÷ 4 = 3
  • 12 ÷ 6 = 2
  • 12 ÷ 12 = 1

Finding common factors

When you want to find the factors of two different numbers, you look at what they have in common. For example, let's find the common factors of 8 and 12.

Factors of 8

  • 1 (since 8 ÷ 1 = 8)
  • 2 (since 8 ÷ 2 = 4)
  • 4 (since 8 ÷ 4 = 2)
  • 8 (since 8 ÷ 8 = 1)

Factors of 12

  • 1 (since 12 ÷ 1 = 12)
  • 2 (since 12 ÷ 2 = 6)
  • 3 (since 12 ÷ 3 = 4)
  • 4 (since 12 ÷ 4 = 3)
  • 6 (since 12 ÷ 6 = 2)
  • 12 (since 12 ÷ 12 = 1)

Now, let's find the common factors of 8 and 12 by comparing both lists:

- Common factors are: 1, 2, 4

Greatest common factor explained

The greatest common factor (GCF) is the largest factor among the common factors of two or more numbers. In the example with 8 and 12:

– The common factors are 1, 2, 4, and the largest of these is 4 Therefore, the GCF of 8 and 12 is 4.

Why is GCF important?

Understanding the GCF is useful for several reasons:

  • This helps in simplifying fractions.
  • This can help solve problems that involve dividing something into equal parts.
  • This lays the groundwork for more advanced math topics like simplifying algebraic expressions and finding the least common multiple.

How to find GCF

There are several methods you can use to find the GCF. Let's look at two common methods: the list method and the prime factorization method.

1. List method

- List the factors of each number. - Identify the common factors. - Choose the largest number from the common factors.

Let us find the GCF of 30 and 45 using the list method.

Factors of 30

  • 1 (since 30 ÷ 1 = 30)
  • 2 (since 30 ÷ 2 = 15)
  • 3 (since 30 ÷ 3 = 10)
  • 5 (since 30 ÷ 5 = 6)
  • 6 (since 30 ÷ 6 = 5)
  • 10 (since 30 ÷ 10 = 3)
  • 15 (since 30 ÷ 15 = 2)
  • 30 (since 30 ÷ 30 = 1)

Factors of 45

  • 1 (since 45 ÷ 1 = 45)
  • 3 (since 45 ÷ 3 = 15)
  • 5 (since 45 ÷ 5 = 9)
  • 9 (since 45 ÷ 9 = 5)
  • 15 (since 45 ÷ 15 = 3)
  • 45 (since 45 ÷ 45 = 1)

The common factors are 1, 3, 5, 15, and the largest of these is 15.

Hence the GCF of 30 and 45 is 15.

2. Prime factorization method

The prime factorization method involves breaking down each number into its prime factors. Prime factors are numbers that are prime (they have only two factors, 1 and themselves). Once we have the prime factors, we can find the prime factors. Identify common prime factors and calculate the GCF.

Example: Find the GCF of 18 and 24

Prime factors of 18

 18 = 2 × 3 × 3

Prime factors of 24

 24 = 2 × 2 × 2 × 3

Common prime factors: 2 and 3

Multiply the lowest powers of all common prime factors: 2 1 × 3 1 = 6.

Therefore, the GCF of 18 and 24 is 6.

Visualizing the GCF

To understand the concept of GCF, consider a simple visual representation using two rectangles:

Imagine you have two rectangles, one with length 18 and the other with length 24:

18 24

The GCF is represented by the largest rectangle that can be used to measure both without leaving a remainder:

6

The largest rectangle that can measure both has length 6, which is the GCF of 18 and 24.

Practice problems

Let's practice finding the greatest common factor with some examples:

  1. Find the GCF of 16 and 24.

    Factors of 16: 1, 2, 4, 8, 16

    Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

    Common factors: 1, 2, 4, 8

    Highest common factor: 8

  2. Find the GCF of 10 and 15.

    Factors of 10: 1, 2, 5, 10

    Factors of 15: 1, 3, 5, 15

    Common factors: 1, 5

    Greatest common factor: 5

  3. Find the GCF of 50 and 75 using prime factorization.

    Prime factors of 50

     50 = 2 × 5 × 5

    Prime factors of 75

     75 = 3 × 5 × 5

    Common prime factors: 5 2

    Highest common factor: 25

Conclusion

The concept of the greatest common factor is fundamental in mathematics and lays the groundwork for more advanced math topics. By understanding how to determine the GCF using methods such as the list method and the prime factorization method, you will enhance your problem-solving abilities. Practice regularly and challenge yourself with different sets of numbers to improve your recognition skills.


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