Breaking down numbers into prime factors

Before moving forward on the definition and other aspects of the Breakdown of numbers into prime factors, it may be useful to review some definitions, which will allow us to understand this mathematical procedure in its precise context.


Fundamental definitions

In this regard, it is also likely to be relevant to delimit this theoretical review into three basic concepts: Natural Number, Prime Numbers, and Composite Numbers, as it is directly related to the nature and categories to which the numerical elements belong on the basis of which the breakdown of numbers into prime factors is performed, a procedure also known as Factorization.  Here are each of these definitions:

Natural numbers

First, it will then be said that Mathematics defines natural numbers as the numeric set made up of all positive integers, i.e. those numbers that range from 1 to the ∞. zero, an element that isn´t considered a number, but which is also understood as part of the Natural Numbers.


On the other hand, these numbers will be understood as the numerical elements thanks to which the elements of a collection can be counted, assigned a number or position, as well as express an accounting amount.

Thus, specialists agree that natural numbers constitute the oldest set of numbers that can be found within Humanity, where they would have been generated from the concept of quantity, and then the elements with which primitive man was able to count and order the world around him.

Prime numbers

For their part, prime numbers would have been defined by Mathematics as all those positive integers – that is, natural numbers – greater than zero (0) and different from one (1) that are specifically characterized by having only two dividers number one (1) and the number itself.

Similarly, the mathematical discipline has indicated that prime numbers will be those numbers that have the quality of not being able to be expressed as products of lower natural numbers. Therefore, the theory says that prime numbers cannot be factored. Likewise, just as the one – by mathematical convention – is neither considered prime nor composite, the two will be the only even number that can be understood as a prime number.

Composite numbers

With respect to the Composite Numbers these will then be understood as those numbers, greater than zero (0) and different to one (1) that have more than two dividers, among which are at least one, the compound number itself and another divider that is between these two extremes:

1 < d < n

Consequently, and unlike Prime Numbers, Composite Numbers will be characterized by their ability to be expressed as the product the powers of prime numbers, i.e. that these numbers may be factored or decomposed in their factors Cousins.

How to break down a number into prime factors

With these definitions in mind, it may certainly be much easier to address the correct process to be followed when decomposing or factoring a number into prime factors. In this order of ideas, it is likely that it is most useful to make this explanation directly in reference to a specific example, such as the one shown below:

Factor or break down the following number into prime factors: 24

The first step to be taken when factoring a number will be to determine whether it is a prime number – which simply cannot be broken down or factored – or a composite number. For this you will need to see if the number in question accepts more than two numbers as dividers. In this case, the number 24 may have the following dividers:

24 : 1= 24
24 : 2= 12
24 : 3 = 8
24: 4 = 6
24: 6 = 4
24 : 24 = 1

In doing so, there will then be evidence that it has more than two dividers, so it will be considered as a Composite Number, which in turn indicates that this number may be broken down into prime factors. For this it will then be necessary to start dividing this number by the lowest value prime numbers:

24       2
12       2
6       2
3       3
1

Therefore, it will then be considered that the number 24, in addition to being able to be identified as a Composite Number, may be broken down into the following prime factors:

24 = 23 . 3

If this result were to be corroborated, it would then be necessary to resolve the following powers and products:

23 . 3 = 8 . 3 = 24

Picture: pixabay.com

Breaking down numbers into prime factors
Source: Education  
September 21, 2019


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