# Gcf and Lcm Worksheet

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GCF and LCM

The Greatest Common Factor - is the largest factor two or more numbers have in common. ***All numbers will
have 1 in common*** There are two methods for finding GCF:

1) List the factors of the numbers then choose the largest

Example: 12 and 18
12 = 1,2,3,4,6,12
18 = 1,2,3,6,9,18

The GCF of 12 and 18 is 6.

2) The other method involves prime numbers and prime factorization.
a) list the prime factors of the numbers using a factor tree
b) If a number appears on both lists, circle both numbers
c) Multiply one of each pair of circled numbers and that is the GCF

Example: 36 and 90
The prime factors of 36 = 2 • 2 • 3 • 3
The prime factors of 90 = 2 • 3 • 3 • 5

Since they have 2, 3, and 3 in common, we multiply 2 • 3 • 3 and the GCF is 18.

****If there are no numbers in common on both lists then the GCF is 1

Multiples

The first 5 multiples of 7 are = 7, 14, 21, 28, 35
You get that by going:
7•1=7                      7 • 3 = 21                7 • 5 = 35
7 •2 = 14                  7 • 4 = 28

The first 6 multiples of 20 are = 20, 40, 60, 80. 100, 120
You get that by going:
20 • 1 = 20               20 • 3 = 60                 20 • 5 = 100
20 • 2 = 40               20 • 4 = 80                 20 • 6 = 120

Least Common Multiple - the smallest non zero multiple that two or more numbers have in common.

There are two methods for finding LCM.

1) write out the multiples of the numbers and choose the smallest multiple in common
that is greater than zero

Example: Find the LCM of 18 and 27

The multiples of 18 are = 18, 36, 54, 72, 90
The multiples of 27 are = 27, 54, 81                            The LCM = 54
2) You can also use prime numbers to find LCM
a) write out the prime factors of both numbers using factor trees
b) write each factor one time
c) write out the greatest number of times that each factor appears when looking at
separate lists

Example: Find the LCM of 18 and 27
The prime numbers of 18 = 2 • 3 • 3
The prime numbers of 27 = 3 • 3 • 3

The factors are 2 and 3.
2 appears most on the list for 18
3 appears most on the list for 27
Multiply 2 • 3 • 3 • 3 = 54
The LCM is 54

Find the LCM of 16, 30, and 27.
16 = 2 • 2 • 2 • 2
30 = 2 • 3 • 5
27 = 3 • 3 • 3
The factors are 2, 3 and 5.
2 appears most on the list for 16 ( 2• 2 • 2 • 2)
3 appears most on the list for 27 (3 • 3 • 3)
5 appears most on the list for 30 (5)
Multiply 2 • 2 • 2 • 2 • 3 • 3 • 3 • 5 =
The LCM is 2160

Lowest Common Denominator

Find the Least Common Multiple of the denominators and you’ve found the Lowest Common Denominator
3      11
Example:      Find the lowest common denominator of          and
15      24
The first step is to find the LCM of 15 and 24 which are the
denominators

15 = 3 • 5
24 = 2 • 2 • 2 • 3
The factors are 2, 3, 5
3 appears most on 15 and 24. (3)
2 appears most on 24 (2 • 2 • 2)
5 appears most on 15 (5)
Multiply 2 • 2 • 2 • 3 • 5 = 120
The Lowest Common Denominator is 120
Factors, Prime Factors, and Multiples

Factors are numbers that divide evenly into another number.

Example: The factors of 20 are --- 1,2,4,5,10,20

Each of the above factors divides evenly into 20.

There are certain easy methods to test divisibility - to see if a number is a factor of another number.

If the number is
divisibly by:                  Test:                   Examples

2             The number ends in 2,4,6,8,0              1,350 or 21,988, or 752

3             Add up all the digits of the number    573 = 15; 3 goes into it evenly
and divide the sum by 3

4              Divide 4 into the last two digits        736 ---> 4 goes into 36 evenly
720 ---> 4 goes into 20 evenly

5              If the number ends in 5 or 0             430 or 3755

6              If both 2 and 3 are factors              780 ends in o so it's divisible
by 2; The sum of the digits is
15 and that's divisible by 3

8              Divide 8 into the last 3 digits           20,504 --- divide 8 into 504.
It goes in evenly so 8 is a factor

9              Add up all the digits of the number       8190 = 18; 9 goes into 18
and see if 9 goes in evenly              evenly so 9 is a factor

If 9 is a factor of a number, 3 must also be a factor BUT if 3 is a factor, 9 is not always a factor***

10             The number will end in 0.                230; 6000; 7540

For numbers 7, 11, 12, 13, 14, 15, 16, 17, etc. you have to actually divide.

Prime numbers and Composite Numbers

Prime Numbers have only two factors - 1 and itself

Example: The factors of 17 are 1 and 17 only. No other number divides in evenly.

Other prime numbers are: 2,3,5,7,11,13,19,23,29,31

Composite numbers have more than 2 factors - 1, itself, and at least 1 other number

Example: 51 - 1, 51 and 3 and 17

To find if a number is prime or composite, use the tests of divisibility.
Practice Worksheet 1

Find all the factors of the following numbers: (Both prime & composite)

1.       26                 2.          40                3.         56             4.       39

State whether each number is divisible by 2,3,4,5,6,8,9,10. List all the possibilities

5.       135                6.          891               7.         5455           8.       3720

9.       414                10.         3870              11.        15,408         12.      1527

Fill in the missing number to make the following divisible by 6: Then make each divisible by 4

13.      402_ 2             14.         71_ 4             15.        1000_

Tell whether each number is prime or composite. If it is composite, tell one other factor that it is
divisible by.

16.      57                 17.         37                18.        56             19.      11,121

20.      63                 21.         117               22.        113            23.      12,543

Using a factor tree, find the prime factorization of each of the following numbers:

24.      210                25.         280               26.        336            27.      1024

28.      415                29.         550               30.        88             31.      67

Multiply:

32.      .006 • .06 =                   33.      4.5 • 65 =                   34.   1.46 • 2.8 =

35.      14.89 • 1.8 =                  36.      .3002 • .7 =                 37.   1.62 • .009 =

38.      21.1 • 6 =                     39.      98 • 1.4 =                   40.   .08 • .0004 =

Divide
41.    24 18               42.         .6.0498          43.         3025         44.      .18.54036

Solve:
n                                                                                   y
45.           + 2 = 3.98          46.    7y - 5 = 30           47.   1.5n = 3.9     48.           - 3 = 48
1.25                                                                                .271
Practice Worksheet 2

Find the GCF and LCM of the following numbers using the factor method:

1.     8 and 14                2.      14 and 21                   3.    24 and 42

Find the GCF and LCM of the following numbers using prime factorization:

4.     28 and 45                       5.         45, 60, 160            6.      32, 80, 120

7.     21 and 28                       8.         18 and 32              9.      70 and 120

10.    20 and 50                       11.        120 and 35             12.     9, 12, and 15

13.    240 and 300                     14.        80 and 180             15.     70 and 160

16.    17 and 53                       17.        175 and 150            18.     135 and 65

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