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Prime Factorization of Three-Digit Numbers

This lesson shows examples on how to get started with the prime factorization of three-digit or larger numbers, using a factor tree. You simply use your knowledge of divisibility tests to get started with the factoring process, and go on from there. The lesson contains many exercises, and you can make even more factoring exercises here.



 

Use the divisibility tests for 2, 3, 4, 5, 6, 9, or 10 in building the factor tree.
  135   
 /    \   
5 × ?

 

  2 7  

5 

)

 1 3 5 
-  1 0
   3 5
   - 3 5
   

0
  135   
 /    \   
 5 × 27    
   /   \
3 × 9   
         /   \  
 3 × 3

We start out by noticing that 135 is divisible
by 5. Long division tells us that 135 = 5 × 27.
Eventually we get 135 = 3 × 3 × 3 × 5.
  441   
/     \
9   ×  ?

  4 9  

9 

)

 4 4 1 
-  3 6
   8 1
   - 8 1
   

0
  441   
/     \
9   ×  49
/    \     /    \
3 × 3 × 7 × 7
 
 

Adding the digits of 441, we get 9, so it is
divisible by 9. We divide to get 441 = 9 × 49.
Eventually, we get 441 = 3 × 3 × 7 × 7.
912
 /       \
4    ×   ?

2 2 8  

4 

)

 9 1 2 
 

-

 8
    1 1
    -  8
        3 2
       - 3 2
    0
   
912
/       \
4  ×  228
  /   \       /    \
2 × 2  × 4 × 57
   
912
/         \  
4     ×   228  
    /  \         /       \   
2 × 2  ×  4    ×   57    
  /   \       /    \   
2 × 2 × 3 × 19

The last two digits
of 912 are “12” so
it is divisible by 4.

228 is also divisible
by 4 (because its last
digits are “28”).

 

Lastly, 57 is factored
to 3 × 19. So, 912 factored
is 2 × 2 × 2 × 2 × 3 × 19.


1. Factor these numbers till the factors are prime numbers. Use a notebook for the long division.

a.  124      

/       \
2  ×   ___
        /      \
b.  260    
 /       \
10   ×   ___
/    \        /    \
c.  96   
 /       \
  3   ×   ___
            /      \

 

 

d.  90 

e.  165  

f.  95  

g.  80

 h.  240 i.  272
j.  76 k.  126 l.  104


2. Factor the following numbers to prime factors. 

a.  196 b.  380 c.  336

d.  306  

e.  116  

f.  720  

g.  675

 h.  990 i.  945

 

Find all primes between 0 and 200. Use the sieve of Eratosthenes again
(you need to make a grid in your notebook). 

This time you need to cross out every 2nd number starting at 4, every 3rd number starting at 6, every 5th number starting at 10, every 7th number starting at 14, every 11th number starting at 22, and every 13th number starting at 26.


 


This lesson is taken from my book Math Mammoth Multiplication & Division 3.

Math Mammoth Multiplication & Division 3

A self-teaching worktext for 5th grade that covers multi-digit multiplication, long division, problem solving, simple equations, ratios, divisibility, and factoring.

Download ($7.40). Also available as a printed copy.

=> Learn more and see the free samples!

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