5.9: Fundamentals 9 - Prime Factorization, GCF, and LCM

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    LEARNING GOALS

    By the end of this lesson, you should understand that

    • it is possible to list all of the factors of a number.
    • a number can always be written as a product of primes (prime factorization).
    • two numbers always have a least common multiple (LCM) and a greatest common factor (GCF).

    By the end of this lesson, you should be able to

    • find all the factors of a number.
    • write the prime factorization of a number.
    • find the greatest common factor (GCF) and least common multiple (LCM) of two numbers using the prime factorization.

    FUNDAMENTALS OF THE LESSON

    A number is called a multiple of another number, if the first number can be written as a repeated addition of the second number. In this case, the first number can be divided by the second number without remainder. For example, the number 6 can be written as 2 + 2 + 2, a repeated addition of 2, and as 3 + 3, a repeated addition of 3. Therefore, 6 is a multiple of 2 and a multiple of 3. We also write 6 = 3 x 2 = 2 x 3. Because the number 6 can be divided by 2 without remainder, we say that 2 is a factor or divisor of 6. We also say that 6 is divisible by 2. The number 3 is also a factor or divisor of 6.

    1. List all of the factors of 12.
    2. List all of the factors of 15.
    3. List all of the factors of 29.
    4. Which is not a factor of 100?

    (i) 1

    (ii) 10

    (iii) 50

    (iv) 25

    (v) 20

    (vi) 30

    TRY THESE 1

    School groups (factors, listing all of the factors of a number)

    1. There are 60 first graders in Forest Hills elementary school. The principal wants to divide them into different groups of the same size. If no students are left out of a group, what is a possible size of the groups in this situation? For example, one answer could be "1," where all students are in a group of their own. What could be another group size?
    2. Mr. Jones and Mrs. Sandoval teach elementary school. There are 20 students in Mr Jones’ class and 24 in Mrs. Sandoval’s class. Each teacher wants to create groups of students in their classes. They would like to have the same size groups in both classes and they would like the groups to be as large as possible. How many students should the teachers put into each group?
    3. Mrs. Singh has 23 students in her class. She wants to divide her students into groups of the same size. What size groups can she make?

    The answer you obtained in question 5 is the list of all divisors (factors) of the number 60. The answer you got in question 6 is the greatest number that divides both 20 and 24 without remainder. It is called the greatest common factor (GCF) of 20 and 24. A prime number is a number that is only divisible (without remainder) by 1 and itself. As you found out in question 7, the number 23 is a prime number.

    1. List all the prime numbers between 1 and 50.

    TRY THESE 2

    Writing as a product (multiplication)

    Mel was given the task of writing the number 6 as a product (multiplication). She wrote 6 = 2 x 3. Factoring a number means writing the number as a product. Mel factored the number 6. She was then asked to factor the number 24 as much as possible. She wrote:

    24 = 4 x 6 = 2 x 2 x 6 = 2 x 2 x 2 x 3.

    Mel realized that the numbers or factors involved in the final multiplication were all prime. She couldn’t factor them anymore without writing a 1. Mel’s final product is called the prime factorization of 24.

    1. Write the prime factorization of the following numbers (Divide the work with the members of your group):

    A. 27

    B. 60

    C. 98

    TRY THESE 3

    Using Prime Factorization to Find the GCF of Two Numbers

    1. Reread question 6 and answer the questions below.

    A. Write the prime factorization of 20 and 24.

    B. Write the prime factorization of your answer to Question 6.

    C. What do you notice in the answers to 10A and 10B?

    D. Now write the prime factorization of the numbers 105 and 140 and find their greatest common factor.

    TRY THESE 4

    Using the Prime Factorization to Find the LCM of Two Numbers

    1. If a number is divisible by 6, should this number be divisible by 3? Explain.
    2. Should the least common multiple of two numbers be divisible by all the factors of both numbers?

    FURTHER APPLICATIONS

    An exponent indicates how many times a number is a multiple. For example, when you see 24, 4 is the exponent. This can be expanded to read 2 x 2 x 2 x 2.

    1. A. Calculate 24 x 3 x 7

    B. Calculate 22 x 33 x 5

    C. Use the prime factorizations from 13A and B to find the greatest common factor of these two numbers.

    14. Use the conclusion you made in question 13 to find the least common multiple of the numbers 24 x 3 x 7 and 22 x 33 x 5.

    Questions: Prime Factorization, GCF, and LCM

    Find the prime factorizations of the following numbers:

    1. 28
    2. 625
    3. 100
    4. 13

    Find the GCF and the LCM of the following pairs of numbers (you can leave your answers in factored form):

    1. 2 x 5 x 11 and 5 x 7
    2. 23 x 3 and 2 x 32
    3. 3 and 4
    4. 4 and 6
    5. 5 and 7
    6. 9 and 45
    7. 12 and 18
    8. 50 and 30

    This page titled 5.9: Fundamentals 9 - Prime Factorization, GCF, and LCM was last modified on Wed, 15 May 2024 07:37:18 GMT and is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Carnegie Math Pathways (WestEd) .