Greatest Common Factors

Getting Started

The student does not have an effective strategy for finding factors of a number.

The student:

• Confuses factors and multiples.
  
     
  
  
• Attempts to use factor trees but breaks numbers down additively.
  
  
  

What does it mean to be a factor of a number? Can you give me an example of a factor of the number six?

What does it mean to be a multiple of a number? Can you give me an example of a multiple of the number six?

How are factors different from multiples?

Review the definitions of factors and multiples emphasizing the difference between them. If necessary, review divisibility rules. Then give the student a number such as 24 and guide the student to test each counting number (up through five, the nearest whole number to the square root of 24) to determine if each is a factor. Have the student list the factors in order and use a “factor rainbow” to complete the list by finding the “factor partners” of the factors already found (i.e., another factor which, along with each identified factor, multiplies to 24). Provide examples of other numbers less than or equal to one hundred and ask the student to systematically find all of the factors of each number. Be sure to include some numbers that are perfect squares.

Review the concept of prime factorization and guide the student to use the prime factorization of a number to find all of its factors. For example, guide the student to rewrite a number such as 24 as 2 x 2 x 2 x 3. Then guide the student to systematically identify all of its factors by first identifying each unique prime factor of 24 (2 and 3), each unique product of two prime factors (2 x 2 and 2 x 3 or 4 and 6), each unique product of three prime factors (2 x 2 x 2 and 2 x 2 x 3 or 8 and 12), and each unique product of four prime factors (2 x 2 x 2 x 3 = 24). Acknowledge that the number one is a factor of every whole number and should always be included in the list of factors.

Review the meaning of greatest common factor and provide the student with examples of two or more numbers along with all of their factors. Guide the student to identify the greatest common factor. Give the student pairs of whole numbers less than 100 and challenge the student to find and list all of the factors of each number and to identify all common factors as well as the greatest common factor. Be sure to include pairs of numbers in which one number is a multiple of the other and in which the numbers are relatively prime (e.g., have no common factor other than one).

Making Progress

The student has an effective strategy for finding factors but errs in identifying the greatest common factor.

The student:

• Does not find all of the factors of each number, including the greatest common factor.
  
     
  
  
• Attempts to use a prime factoring strategy but makes errors in determining the greatest common factor. For example, the student rewrites 24 as 2 x 2 x 2 x 3 and 36 as 2 x 2 x 3 x 3 but says the greatest common factor is 2 x 3 = 6.
  

How do you know if you have found all of the factors of a number?

What does it mean to be a common factor of two numbers?

What does it mean to be the greatest common factor of two numbers?

Review strategies for finding factors of numbers and encourage the student to be systematic and thorough regardless of strategy. Provide the student with tips to help ensure that all factors are found, such as:

• Use division to find all factors less than or equal to the square root of the number and then use a “factor rainbow” to identify the remaining factors.
• Always include the number one and the number itself in the list of factors.
• List factors in order from least to greatest.

Encourage the student to use a strategy based on prime factorization for finding the greatest common factor. For example, have the student rewrite 24 as 2 x 2 x 2 x 3 and 36 as 2 x 2 x 3 x 3. Then show the student that the factors of the greatest common factor are the common prime factors (i.e., 2 x 2 x 3 = 12). This method, if implemented correctly, ensures identifying the greatest common factor without having to find all of the factors of each number.

Give the student additional pairs of whole numbers less than 100 and challenge the student to find and list all of the factors of each number and to identify all common factors as well as the greatest common factor. Be sure to include pairs of numbers in which one number is a factor of the other and in which the numbers are relatively prime (i.e., have no common factor other than one).

Got It

The student provides complete and correct responses to all components of the task.

The student lists all factors of each number and correctly identifies the greatest common factor.

What is the difference between factors and multiples?

What strategy did you use to find the factors of each number?

Are there any other strategies that you can use to find factors and the greatest common factor?

Can two numbers have more than one common factor? Can they have more than one greatest common factor?

What is the least common factor of any set of whole numbers?

If the student is not already doing so, encourage the student to use a strategy based on prime factorization for finding the greatest common factor. For example, have the student rewrite 24 as 2 x 2 x 2 x 3 and 36 as 2 x 2 x 3 x 3. Ask the student to use exponents to express the factorizations as  and . Then show the student that the greatest common factor is the product of each unique prime factor raised to the lowest power found in the prime factorizations (i.e., ).

Give the student additional pairs of whole numbers less than 100 and challenge the student to use this strategy to find the greatest common factor. Be sure to include pairs of numbers in which one number is a multiple of the other and in which the numbers are relatively prime (i.e., have no common factor other than one).

Have the student consider when each approach (listing factors versus using prime factorizations) to finding the greatest common factor might be most appropriate.