As you may already know, a number can have a lot of factors. A bunch of factors multiplied together to represent a number is called factorization. Well prime factorization just a unique way of showing the factorization of a number-by reducing the number into a series of only prime numbers multiplied together.
Let's take the number By convention, the prime factors are written in numerical order and common factors are expressed as exponents.
One way is to divide the number by prime factors until you are left with a prime number. Personally, I start dividing by the lowest prime factor each time and work my way up. Instead of me explaining through text all the time, let us do one example. We have the number To find the prime factorization of 60 we start off by looking at the smallest prime 2.
Now we have After checking 2 again, we find that 2 is no longer a factor so we go onto the next prime 3. Before we begin with prime factorization let's review the concepts of prime numbers and factoring. Factoring is a way to break a number down by finding smaller numbers that can be multiplied together to get that number.
Prime Numbers are numbers that only have two factors, one and itself i. When we are asked to find the prime factorization of a number we are being asked to break a composite number down into only prime numbers. This is done by finding a factor pair of the given composite number and then factoring that factor pair until we are left with only prime numbers. I like to use a visual called a factor tree to help me find all of the prime factors of a number.
Below, I used a factor tree to find the prime factorization of For example, prime factorization of 40 is the representation of 40 as a product of prime numbers and can be done in the following way:. Let us see the prime factorization chart of a few more numbers in the table given below:.
Prime factorization is similar to factoring a number and considering only the prime numbers 2, 3, 5, 7, 11, 13, 17, 19, and so on among all the factors. The factors are the numbers that divide the original number completely and can't be split into more factors are known as the prime factors.
Factors of a number are the numbers that are multiplied to get the original number. For example; 4 and 5 are the factors of 20, i. For example: 2, 2, and 5 are the prime factors of 20, i.
There are various methods to find the prime factorization of a number. The most common methods used to find the prime factorization are given below:. In the factor tree method, the factors of a number are found and then those numbers are further factorized until we reach the prime numbers. To evaluate the prime factorization of a number using the factor tree method, follow the steps given below:.
Example: Follow the diagram given below to understand the concept and find the prime factorization of The division method can also be used to find the prime factors of a large number by dividing the number by prime numbers.
Follow the steps given below to find the prime factors of a number by using the division method:. There is a wide range of properties of prime factorization. The two most important applications of the prime factorization are given below. Cryptography is a method of protecting information and communicating cryptography through the use of codes. Prime factorization plays an important role for the coders who want to create a unique code using numbers that is not too heavy for computers to store or process quickly.
For this, we first find the prime factorization of both the numbers. Next, we consider the following:. We could keep going. I think you get the general idea. You move to 7, 7 is prime. It's only divisible by 1 and 7. Prime is not the same thing as odd numbers. Then if you move to 10, 10 is also not prime, divisible by 2 and 5.
And we could keep going on like this. People have written computer programs looking for the highest prime and all of that. So now that we know what a prime is, a prime factorization is breaking up a number, like 75, into a product of prime numbers. So let's try to do that. So we're going to start with 75, and I'm going to do it using what we call a factorization tree.
So we first try to find just the smallest prime number that will go into Now, the smallest prime number is 2. Does 2 go into 75? Well, 75 is an odd number, or the number in the ones place, this 5, is an odd number. But 6 is not a prime number, so we need to go further. Example: What is the prime factorization of ? Can we divide exactly by 2? The next prime, 5, does not work.
Example: What is the prime factorization of 17?
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