Factors and Multiples – How to Find the GCF and LCM

By Stuart Ackerman

     It is essential for students to have a good understanding of the greatest common factor (GCF) and the lowest common multiple (LCM) when learning about fractions.

The Greatest Common Factor

     The greatest common factor (GCF) is the largest common factor of two or more numbers.  The GCF can be used to simplify a fraction.  We can use Prime Factorization to find the GCF of numbers.

     A great way to find the greatest common factor is to first find the factors of two numbers and determine the largest number they have in common.

     For example, when we carry out prime factorization of the numbers 45 and 60, we find the factors of 45 are 3, 3, and 5 (i.e. 3 x 3 x 5 = 45).  The factors of 60 are 2, 2, 3, and 5 (i.e. 2 x 2 x 3 x 5 = 60).  What factors do they have in common?  3 and 5.  So we multiply these common factors, 3 and 5 to get 15. 

     Therefore the answer is 15.

     What did we do?

1.    We found the factors of 45 and 60 (you should know prime factorization).

2.    We looked for common factors (i.e. 3 and 5).

3.    Then we multiplied 3 x 5 = 15.

     Another way to find the greatest common factor is to list all the factors of both numbers and find the largest factor that they both have in their list.

The Least Common Multiple

     The least common multiple (LCM) is the smallest number that two or more numbers will divide into evenly.  It is the smallest number that is a multiple of both.  The LCM of two or more numbers is used when adding or subtracting fractions.  Zero cannot be a common multiple.

Let’s find the LCM for 3 and 4.

The multiples for 3 are:  3, 6, 9, 12, 15, 18, 21, 24

The multiples for 4 are:  4, 8, 12, 16, 20, 24

Clearly, 12 is the lowest common multiple because it is the smallest number that is a multiple of both.

 ©Tutorgiant.com