Just as words can be broken down by syllables, numbers can be broken down into smaller components. These components are called factors. The smallest factor of a number is called the prime factor.
Definition of a Prime Number
Prime numbers are numbers which can only be divided by 1 or the prime number itself. Examples of prime numbers include: 2, 3, 5, 7, 11, and 13.
Numbers like 4, 6, and 8 are not prime numbers as they are divisible (able to be divided) by other numbers such as 2 (4=2*2), 3 (6=2*3), and 4 (8=2*4).
What is Prime Factorization?
Prime factorization is taking a large number and breaking it down into prime factors. It takes a larger integer (number) and finds the smaller numbers which when multipled together result in the larger number.
The prime factorization of 13 is 1 times 13. It cannot be simplified any further.
The prime factorization of the number 9, however, could be broken down into smaller parts.
9 is equal to 9 times 1, yet the number 9 is not prime.
Nine can be factored into 3 times 3. Since 3 is a prime number, the prime factorization of 9 is 3 times 3. It cannot be factored any further.
Example of Finding Prime FactorsThe prime factorization of 72 takes a little longer, but the same rules apply as what was done with the prime factorization of 9.
The first step would be to take the small prime number of 2 and see if 72 can be divided by 2 evenly without a remainder. This would by definition make 2 a factor of 72.
72/2= 36
Therefore, the prime factorization of 72 looks like this so far:
2*36
However, 36 is not yet prime. 36 is also divisible by 2 (36/2=18).
The prime factorization of 72 now reads as:
2*2*18
18 equals 2 times 9.
This is what the prime factorization looks like at this step:
2*2*2*9
As shown above, 9 is made up of 3 times 3.
The final prime factorization step is to break the 9 down into 3 times 3.
The result is the prime factorization of 72= 2*2*2*3*3 or (2^3)*(3^2)
The way one knows the prime factorization is that all the factors are prime numbers.
Another Example of Prime Factorization
The prime factorizations where the numbers can be broken down by 2 or 3 are easy, but what about numbers which have bigger prime factors?
Here is an example. What is the prime factorization of 121?
Clearly, 121 is not divisible by 2, as 121 is odd not even, and also ends with 1.
What about 3? It is not possible to have 121 evenly divisible by 3, since 120 is. It would have a remainder of 1. Therefore, 3 does not factor out either.
The same holds true when 5 is tried; if only the number was 120 not 121.
Ultimately, 11 is found to work. This makes sense, since 110 is the product of 11*10, so 11*11 must equal 121. After all, 110+ one more set of 11 = 121.
The prime factorization is thereby found to be:
121= 11*11.
This shows that sometimes the smaller prime numbers do not factor out of the number, so larger prime numbers must be tried. One cannot just assume that 121 is a prime number. Familiarizing one's self with common multiples like 110=11*10 will help speed this process.
In closing, this is how prime factors are found, so that they may be used in finding common denominators or reducing fractions into simplest form.
 |
