Notes on Greatest Common Factor and Least Common Multiple

Factors are the numbers that multiply together to get another number.

A Product is the number produced by multiplying two factors.

To factor a number

 Write 1 and the number itself separated by some space.  Test the number for divisibility by 2. If it is, write 2 and the other number inside the first two.  Continue testing and writing the factor pairs inside the previous pair.  When you reach the middle, you are finished.

Example: Another way to get the factors of a number is to list 36 { 1, 36 } its factor pairs using a systematic approach:

36 { 1, 2, 18, 36 } 1 x 36, 2 x 18, 3 x 12, 4 x 9, 6 x 6

36 { 1, 2, 3, 12, 18, 36 } So the factors of 36 are:

36 { 1, 2, 3, 4, 9, 12, 18, 36 } 1, 2, 3, 4, 6, 9, 12, 18, 36

36 { 1, 2, 3, 4, 6, 9, 12, 18, 36 }

In the example above, 5 is not a factor of 36 so it is not in the factor list. 6 is in the middle so stop there, because 7 is not a factor of 36, nor is 8. The next number, 9, is already in the list, and every number greater than 9 has been included or eliminated when the lower factors were added.

To find the greatest common factor (GCF) of two numbers using the list method

 List the factors for each number as described above

36 { 1, 2, 3, 4, 6, 9, 12, 18, 36 }

54 { 1, 2, 3, 6, 9, 18, 27, 54 )

 Circle the factors in both lists (common to both).

36 { 1, 2, 3, 4, 6, 9, 12, 18, 36 }

54 { 1, 2, 3, 6, 9, 18, 27, 54 )

 The factors you circled are the common factors. The largest of these is the GCF.

1, 2, 3, 6, 9, 18 18 is the GCF of 36 and 54.

To find the greatest common factor with the upside down division bracket

 Write the two numbers inside a bracket, as shown.

36 54

 Divide by the smallest prime number that will divide into both numbers, often 2 or 3.

2 36 54

18 27

 If the quotients (answers) on bottom have a common prime factor, repeat the first 2 steps.

2 36 54

3 18 27 Note: 2 will not divide into 27, but 3 divides into both.

6 9

 Continue this process until the only common factor is 1.

2 36 54

3 18 27

3 6 9

2 3

 Draw a large G (for GCF) around the numbers on the left outside the bracket. Multiply these numbers together, and the product is the GCF of the two numbers.

2 36 54 2 x 3 x 3

3 18 27 6 x 3

3 6 9 18 18 is the GCF of 36 and 54.

2 3

Multiples are the products of a number and another whole number (x1, x2, x3, . . . ).

To find a list of the multiples of a number

 Write the number. That is the number times 1. Then write the number x2, x3, and so on.

4, 8, 12, 16, 20, 24, 28 . . . This is a list of the multiples of 4 up to 28.

To find the least common multiple (LCM) of two numbers with the list method

 List the multiples for both numbers until you discover a number that is in both lists.

4, 8, 12, 16, 20, 24, 28 . . .

7, 14, 21, 28 . . .

 The first and smallest number that is in both lists is the LCM.

The LCM of 4 and 7 is 28.

To find least common multiple (LCM) of two numbers with the upside down division bracket

 Write the two numbers inside a bracket, as shown.

36 54

 Divide by the smallest common prime factor, continuing until the only common factor is 1. This is the same process used in finding the GCF (see notes above for details). Encircle the numbers on the left AND bottom with a large L (for LCM). Multiply the numbers you circled. The product is the LCM of the two numbers.

2 36 54 2 x 3 x 3 x 2 x 3

3 18 27 6 x 3 x 2 x 3

3 6 9 18 x 2 x 3

2 3 36 x 3

108 The LCM of 36 and 54 is 108.