Greatest Common Factor

GCF is the greatest common factor of two or more numbers. In other words, GCF of two numbers is the largest number that divides the given numbers without leaving a remainder. There are different methods to find the GCF of any number, like listing out the common factors, prime factorization, and the division method. 
 

Factors are numbers that divide the given number evenly. For instance, the factors of 12 are 1, 2, 3, 4, 6, and 12. Common factors are the factors that are common in any two or more numbers. For example, the common factors of 12 and 18 are 1, 2, 3, and 6.

GCF and LCM are the two fundamental concepts in mathematics. Now, let’s discuss the difference between LCM and GCF. 

In this section, let’s see how to find the GCF of any two or more numbers. The common methods used to find the GCF are:

• Listing Out Common Factors
   
• Prime Factorization
   
• Division Method

Listing Out Common Factors

In this method, we write down the factors of the given number and identify the common factors to find the GCF. Follow these steps to find the GCF of any two or more numbers using listing out common factors:
 

Step 1: Write down all the factors of the given number.

Step 2: Identifying the common factors from the list. 

Step 3: The largest number from the common factors is the GCF.  

Now, let’s find the GCF of 81 and 72.

Step 1: List the factors of 81 and 72.

The factors of 81 are 1, 3, 9, 27, 81.
The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.

Step 2: Identify the common factors from the list. 

The common factors of 81 and 72 are 1, 3, 9.

Step 3: Find the largest number among the common factors

Here, the largest common factor is 9.

So, the GCF of 81 and 72 is 9.

Prime Factorization

In this method, the given number is broken down into its prime factors. Then the common prime factors are identified and multiplied to determine the GCF. Follow the steps mentioned below to find the GCF: 

Step 1: Breaking the given number into its prime factors

Step 2: Identify the common prime 

Step 3: Multiplying the common prime factors 

Now, let’s find the GCF of 28 and 24.

Step 1: Breaking the given number into its prime factors

The prime factorization of 28 is 2 × 2 × 7 = 2² × 7.

The prime factorization of 24 is 2 × 2 × 2 × 3 = 2³ × 3.

Step 2: Identifying the common prime.

The common prime factor is 2

Step 3: Take the smallest power of the common factor.

Smallest power of 2 = 2² = 4.

GCF of 28 and 24 is 4. 

Division Method

In this method, we divide the larger number by its smaller counterpart. Then the remainder is divided by the previous divisor. The process is repeated until the remainder is zero. Follow the steps given below to find the GCF using the division method:

Step 1: Dividing the largest number by the smallest number

Step 2: If the remainder is 0 in the first step, then the divisor in step 1 will be the dividend in step 2. Now, the remainder will be the new divisor.

 
Step 3: When the remainder becomes 0, the divisor will be the GCF. If not, the process will be repeated.

Now, let’s find the GCF of 54 and 42.

Step 1: Dividing the largest number by the smallest number. So dividing 54 by 42, then the remainder is 12. 

Step 2: Now divide the previous divisor 42 by the remainder 12.
42 ÷ 12 = 3 remainder 6. 

Step 3: Divide the previous divisor 12 by the new remainder 6.
12 ÷ 6 = 2 remainder 0.


Step 4: When the remainder becomes 0, the divisor is the GCF. Therefor, the GCF of 54 and 42 is 6. 

Understanding the Greatest Common Factor (GCF) improves one's number sense and problem-solving abilities which are necessary for proficient arithmetic skills and everyday use. Here are some tips to provide support and strategies while building the foundation:

• Use prime factorization: This is a quick way to break the numbers into prime factors and obtain common factors for the GCF.
  
   
• Use Euclid’s algorithm: Repeatedly divide to find the GCF of any two numbers, especially larger numbers, which can be done readily.
  
   
• List and compare: Listing the factors of the smaller numbers and finding the greatest common factor for smaller numbers, if used, is straightforward.
  
   
• Verify with multiples: The greatest common factor can be verified by comparing to find out if it divides all the numbers or no number greater divides all the numbers.
  
   
• Practice word problems: Practice using real life situations related to sharing or grouping or measuring.
  
   
• Connect GCF to real-life examples: Relate GCF to everyday activities like dividing snacks equally, grouping students, or cutting ribbons into equal parts. This helps students visualize the concept of common factors in real situations.
  
   
• Use visual aids: Employ factor trees, arrays, or Venn diagrams to illustrate how factors overlap between two numbers. Visual learning strengthens understanding of what the GCF is and how to find the GCF.
  
   
• Interactive tools and GCF calculators: Encourage the use of a GCF calculator or online games to quickly check answers. This helps students verify their manual calculations and build confidence.
  
   
• Use skip counting for younger students: For smaller numbers, show how listing multiples and identifying common ones can lead to the GCF. It is a simple approach to teach students.
  
   
• Promote peer teaching: Pair students to solve GCF problems together. Teaching peers helps reinforce their own understanding and improves retention.

The Greatest Common Factor (GCF), also referred to as the Highest Common Factor (HCF), is defined as the largest number that can evenly divide two or more integers without a remainder, and it is useful for simplifying mathematical work. The usefulness of GCF goes beyond mathematics, with other practical applications in decision-making and efficiency. 
 

• Cooking Portions: To ensure that there are no leftovers, and less food waste is created while using the greatest equal portions of ingredients, GCF is can be used in the preparation of meals and serving sizes.
   
• Gardening and Landscaping: The optimal equal spacing of plants can be determined by the greatest common factor. While still allowing the plant to grow, GCF allows for efficient use of available land for growing plants without overcrowding.
   
• Packing Boxes: Businesses can maximize space and minimize empty space by using the GCF to find the largest box size that will fit all products within the same box.
   
• Fencing a Field: Farmers can use the GCF to establish fencing equally around the field at the greatest distance, allowing for the best material costs while maintaining a strong fence that is uniform.
   
• Crafting / Cutting Material: When creating or cutting a material (ex. fabric or wood), the GCF can be used to create or cut the material into the largest equal pieces to minimize waste and maximize efficiency in production.