Prime Factorization of a Number
Prime factorization is the process of expressing a composite number as a product of its prime factors. This is done by dividing the number repeatedly by prime numbers until the final quotient is 1. Each number has a unique prime factorization, regardless of the order of the factors.
In arithmetic, numbers are classified as either prime or composite based on their divisibility properties.
A number is called a prime number if it can only be divided by 1 and itself.
For instance, 2, 3, 5, and 7 are all prime numbers.
Numbers that are neither prime nor 1 are known as composite numbers because they can be expressed as a product of prime numbers—a process known as prime factorization.
For example, 4 is a composite number because it can be divided by 2, in addition to 1 and itself.
Breaking down composite numbers into prime factors is one of the fundamental concepts in mathematics and is often considered a basic “building block” of number theory.
Examples of Prime Factorization
To make this clearer, let’s walk through an example of prime factorization.
If you try to factorize 12, you can express it as:
$$ 12 = 2 \cdot 2 \cdot 3 = 2^2 \cdot 3 $$
In this case, all factors (2 and 3) are prime numbers, so 12 has been correctly factorized into primes.
How to Perform Prime Factorization
To factorize a number \( N \) into primes, follow these steps:
- Start by dividing \( N \) by prime numbers, beginning with the smallest primes (2, then 3, 5, 7, etc.). Keep dividing by the same number as long as the quotient remains a whole number greater than one.
If \( N \) ends in zero (e.g., 360), you can start by dividing it by \( 2 \cdot 5 \), or 10, until the number no longer ends in zero. Then continue dividing by 2, 3, 5, and so on.
- Move to the next prime number only when the quotient is no longer divisible by the current prime.
- Continue dividing until the final quotient is 1. The factorization process ends when the quotient is 1.
At this point, the prime numbers you used to divide \( N \) are its prime factors. The final factorization will be the product of these factors.
Factorization Example
Let’s go through an example with the number 36.
Start by dividing 36 by 2. The quotient of 36 divided by 2 is 18.
$$ 36 : 2 = 18 $$
The number 18 can still be divided by 2.
$$ 18 : 2 = 9 $$
The number 9 isn’t divisible by 2 but can be divided by 3.
$$ 9 : 3 = 3 $$
And 3 can be divided by 3 again.
$$ 3 : 3 = 1 $$
Now that the quotient is 1, the process is complete.
Thus, the prime factorization of 36 is as follows:
$$ 36 = 2 \cdot 2 \cdot 3 \cdot 3 = 2^2 \cdot 3^2 $$
Now, all the factors are prime numbers.
When factorizing, it’s helpful to draw a line and write the starting number on the left, with the first prime divisor to the right. Then, write the quotient below and repeat the process with the new number until reaching 1. The numbers on the right of the line represent the prime factors of the starting number. For example, in the case of 36, you can write: $$ \begin{array}{r|l} 36 & 2 \\ 18 & 2 \\ 9 & 3 \\ 3 & 3 \\ 1 & \\ \end{array} $$ This technique helps keep track of the steps in an orderly way, making the factorization clearer and easier to follow.
Another Example
Prime factorization can be more complex for larger numbers, but the approach stays the same.
Consider, for example, the factorization of 720:
This number ends in zero, so you can initially divide it by \( 2 \cdot 5 \), obtaining 72 as the quotient.
$$ \begin{array}{r|l} 720 & 2 \cdot 5 \\ 72 & \\ \end{array} $$
After this initial simplification, you can start dividing by prime numbers from 2 onward.
Divide 72 by 2 (the smallest prime) and write the quotient in the next row:
$$ \begin{array}{r|l} 720 & 2 \cdot 5 \\ 72 & 2 \\ 36 & \\ \end{array} $$
36 is still divisible by 2.
$$ \begin{array}{r|l} 720 & 2 \cdot 5 \\ 72 & 2 \\ 36 & 2 \\ 18 & \\ \end{array} $$
18 is also divisible by 2.
$$ \begin{array}{r|l} 720 & 2 \cdot 5 \\ 72 & 2 \\ 36 & 2 \\ 18 & 2 \\ 9 & \\ \end{array} $$
The number 9 is odd, so it’s not divisible by 2.
Move to the next prime and divide by 3.
$$ \begin{array}{r|l} 720 & 2 \cdot 5 \\ 72 & 2 \\ 36 & 2 \\ 18 & 2 \\ 9 & 3 \\ 3 & \\ \end{array} $$
And 3 can be divided by 3 again.
$$ \begin{array}{r|l} 720 & 2 \cdot 5 \\ 72 & 2 \\ 36 & 2 \\ 18 & 2 \\ 9 & 3 \\ 3 & 3 \\ 1 & \\ \end{array} $$
The process ends here as the result is 1.
Each number to the right of the line represents a prime factor of the starting number.
Thus, the prime factorization of \(720\) is:
$$ 720 = 2 \cdot 5 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 $$
$$ 720 = 2^4 \cdot 3^2 \cdot 5 $$
This is the product of the prime factors of \(720\).
This factorization shows that every composite number can be uniquely represented as a product of primes.
This principle is formalized in the Fundamental Theorem of Arithmetic, which states that each composite number has a unique prime factorization, apart from the order of the factors. For example, 15 can only be expressed as \(3 \cdot 5\) or \(5 \cdot 3\); no other combination of primes produces the same result.
In conclusion, understanding how to break down numbers into prime factors allows you to explore the underlying structure of numbers and use this knowledge to simplify calculations.
