Prime factorization


Before learning about prime factorization, please read prime and composite numbers.


What is prime factorization?


It's the expression of a number as a product of its prime factors.


For each step in the factorization process, start with the smallest possible prime which divides evenly into the number being factorized.


For instance, consider the prime factorization of 4


The smallest possible prime which will divide evenly into 4 is 2.


Now 4 = 2 x 2 - note how you end up with a second prime (2 again).


You can't factorize any further, so the prime factorization of 4 is expressed

4   =   2  x  2


What about 8 as a product of its primes?


The smallest prime which divides evenly into 8 is 2.


So 8 = 2 x 4.


Next consider the smallest prime which will divide evenly into 4. We know this is 2 from the first example, therefore 8 = 2 x 2 x 2.


Again, you can't factorize further, so 8 as a product of its primes is expressed

8   =   2  x  2  x  2


Some other examples


See if you can follow the factorization steps in each case


  • 6 = 2 x 3
  • 9 = 3 x 3
  • 10 = 2 x 5
  • 12 = 2 x 6 = 2 x 2 x 3
  • 14 = 2 x 7
  • 15 = 3 x 5
  • 16 = 2 x 8 = 2 x 2 x 4 = 2 x 2 x 2 x 2
  • 18 = 2 x 9 = 2 x 3 x 3
  • 20 = 2 x 10 = 2 x 2 x 5
  • 21 = 3 x 7
  • 24 = 2 x 12 = 2 x 2 x 6 = 2 x 2 x 2 x 3
  • 25 = 5 x 5
  • 27 = 3 x 9 = 3 x 3 x 3
  • 28 = 2 x 14 = 2 x 2 x 7
  • 30 = 2 x 15 = 2 x 3 x 5
  • 32 = 2 x 16 = 2 x 2 x 8 = 2 x 2 x 2 x 4 = 2 x 2 x 2 x 2 x 2
  • 35 = 5 x 7
  • 36 = 2 x 18 = 2 x 2 x 9 = 2 x 2 x 3 x 3
  • 40 = 2 x 20 = 2 x 2 x 10 = 2 x 2 x 2 x 5
  • 42 = 2 x 21 = 2 x 3 x 7


Factor tree


This is a similar method only you end up with a 'tree' with the prime factors hanging on 'leaves' off the 'branches'.


For instance, here's a factor tree for the prime factorization of 48 - the prime factors are highlighted



i.e. 48 = 2 x 24 = 2 x 2 x 12 = 2 x 2 x 2 x 6 = 2 x 2 x 2 x 2 x 3


How about a factor tree for the prime factorization of 72?



i.e. 72 = 2 x 36 = 2 x 2 x 18 = 2 x 2 x 2 x 9 = 2 x 2 x 2 x 3 x 3


Over to you!


Try making your own factor trees using the figures from the previous examples (factorization of 6, 9, 10, 12, 14, 15 etc).


Place the number being factorized on top of the tree and branch out to find its prime factors!


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