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The ideas in this lesson are taken from Multiplication Division 3 ebook. Only a few examples of each problem type are shown; you should make more problems of each kind for the student.


Factoring and prime numbers
Free lesson plan from HomeschoolMath.net

This 'lesson' should actually be divided into several lessons for the student.

In any multiplication the numbers that are multiplied are called factors and the result is  called a  product.
 
factor factor product
7 × 6 = 42

From this multiplication fact we can make two division facts:
42 × 6 = 7   and    42 × 7 = 6.
Therefore, 42 is divisible by both 6 and 7.  We say 6 and 7 are divisors of 42.


Example problems

1.  Express each number as a product of two factors.  Sometimes there are several ways of doing this and you can choose which way you like, but don't use 1 × the number itself.  Look at the example.  This process is called factoring.

product factors    product factors    product factors   

10 =

50 =

24 =

5 × 2      

12 =

22 =

110 =

          

94 =

121 =

40 =

2.  Express each number as a product of two factors and then make division facts and state by which numbers it is divisible.  Look at the examples.  Sometimes there are many ways that you can express a number as a product of two factors.

Products: Division facts:
10 = 5 × 2
 

10 = 10 × 1
 

     10 × 5 = 2
10 × 2 = 5

10 × 10 = 1
10 × 1 = 10

  

10 is divisible by 1,2, 5, and 10.

Products: Division facts:
20 = 5 × 4
 

20 = 10 × 2
 

20 = 20 × 1
 

     20 × 5 = 4
20 × 4 = 5

20 × 10 = 2
20 × 2 = 10

20 × 20 = 1
20 × 1 = 20

 20 is divisible by 1, 2, 4, 5, 10, and 20.
Products: Division facts:
15 = __ × __
 

15 = __ × 1
  

  
  

15 is divisible by ________________.

Products: Division facts:
14 = __ × __
 

14 = __ × 1
 

  
 

 14 is divisible by ________________.

Products: Division facts:

11 = __ × 1
 

  
  

 

 

11 is divisible by ________________.

Products: Division facts:
18 = __ × __
 

18 = __ × __
 

18 = __ × 1
 

  
 18 is divisible by ________________.

3.  Use the previous exercise as a help and make a divisibility list of  numbers between 1 and 20.  Circle every number that is divisible only by 1 and the number itself.  Such numbers are called  prime numbers.  (Usually 1 is excluded and is not counted as a prime number.)

number divisible by:
(divisors)
number

divisible by:
(divisors)

number

divisible by:
(divisors)

number

divisible by:
(divisors)

1 1 6   11   16  
2   7   12   17  
5   10

1, 2, 5, 10

15   20

 

 

factor factor product
1 × 11 = 11

Some numbers only have two divisors: 1 and the number itself. Such numbers are called prime numbers.  11 is one of them.

Earlier we found that prime numbers less than 20 are

2, 3, 5, 7, 11, 13, 17,  and 19.

1 is usually not counted as a prime number.

 

product      24
/  \
factors 4 × 6
     
24
/     \
4 × 6
/  \ /  \
2 × 2 × 2 × 3
Here 24 is factored to 4 × 6. We can continue by factoring 4 and 6. We cannot factor 2 or 3 any further (except to 2 = 2 × 1 and 3 = 3 × 1) because they are prime numbers. So 24 = 2 × 2 × 2 × 3.
     30
/   \
5 × 6

  /    /   \

5 × 2 × 3
    We cannot factor 5 any further because it is a prime number. So 30 = 5 × 2 × 3.

21

/  \
3 × 7
    Both 3 and 7 are prime numbers, so we cannot factor any further. So 21 = 3 × 7.

Example problems

1.  Fill the table and circle each prime number between 20 and 30. 

number divisible by:
(divisors)
number

divisible by:
(divisors)

21   26  
22   27  
23   28  
24   29  
25   30  

 

2.  Factor the following numbers so the factors are prime numbers.

18
/ \

 

 

 

6
/ \

 

 

 

14
/ \

 

 

 

25
/ \

 

 

 

 

33
/ \

 

 

 

 

15
/ \

 

 

 

 


 

Prime numbers are like building blocks of all numbers.  They are the first and foremost, and other numbers are 'built' from them.  ALL numbers can be factored down so the factors are just prime numbers.  That is sort of amazing - think about it!
For example, try these bigger numbers:
 
100
/     \
10  × 10
/   \     /   \

          ×         
56
/ \

 

 

 

64
/ \

 

 

 

84
/ \

 

 

 

Does it matter how you start your factorization?  Try and see.  
Does it make any difference if you start out by factoring 24 into 6 × 4,  8 × 3  or  12 × 2?

24
/     \
6 × 4
/ \ / \
                       
24
/     \
8 × 3
/ \ |
                       
24
/     \
12 × 2
/ \ |

 

 

 

We can do the factoring process backwards, too.  Let's start with the building blocks and build numbers:
 
2 × 5 × 2 × 2

\ /         \ /

10 × 4
\         /
40
 
2 × 3  ×  2 × 3  ×  2
\ / \ /

 |

6 × × 2
|

      \      /

×  
\             /     

           72

   
2 × 7  ×  2 × 3
\ / \   /
×
    \          /     

      

 
3.  Build numbers from primes.

2 × 5  × 11
 \    /        |
3 × 2 × 2 × 2
  \   /       \   /

2 × 3  × 7
 \   /        |

 

 

 

 

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