Prime factorization introduction
Prime factorization otherwise known as integer factorization of a number can be defined as the determination of the set of prime numbers that can be multiplied together to give the original integer. A prime number also referred to simply as prime are any natural number that is greater than 1 but has no positive divisors other than 1 as well as itself. A natural number that is more than 1 and is not a prime number is known to be a composite number. E.g. 5 is said to be prime because 1 and 5 are its only positive factor integers, whereas 6 is composite because it has the divisors 2 and 3 in addition to 1 and 6. Given any positive integer n>=2, the prime factorization can be written in the format as below:
- n = α1 α2 p1 p2 …αk pk
- Where the p1 represents the k prime factors, each of order α1, and factor α1 p1 are known as primary. Prime factorization can be performed in the Wolfram language with the use of the command Factor integer [n], that will return a list of p1 as well as α1 pairs. Through him inventing the Pratt certificate, Pratt (1975) became the first to demonstrate that prime factorization lies in the complexity class NP (i.e. non-deterministic polynomial time)
The Wolfram language is designed for the new generation of programmers which has a vast depth of algorithms that are said to be built-in and knowledge, all automatically accessible via its stylish and unified symbolic language. Scalable for programs from small to big with immediate deployment in the cloud and locally, the Wolfram language develops on principles that are clear and the ability to create what promises to be the world’s most productive programming language
The first few prime factorizations i.e. from 1 to 20 whose prime factorization equals to 1 are given in the table below:
- n prime factorization
1 1
2 2
3 3
4 2^2
5 5
6 2·3
7 7
8 2^3
9 3^2
10 2·5
11 11
12 2^2·3
13 13
14 2·7
15 3·5
16 2^4
17 17
18 2·3^2
19 19
20 2^2·5
- Notice that in the above prime factorization, some of the factor integer n numbers were still written the same way as the original integer. This is so because they are only natural prime numbers whereas those integers whose prime factorization were conveyed in different formats as to its original factor integers are known as the composite numbers
Different methods used in prime factorization with solved examples
Prime factorization is a way of determining which from any of the naturally occurring prime numbers will give back its original number when multiplied together with another number. There are several methods that can be used to determine the prime factorization of any number. One of such ways is to basically to divide the original number by prime numbers in a repeated manner as shown in the examples below:
-
- Example 1: Prime factorization of 96 using division method will be to continuously divide 96 by a prime number it is no more divisible:
96 ÷ 2 = 48
48 ÷ 2 = 24
24 ÷ 2 = 12
12 ÷ 2 = 6
6 ÷ 2 = 3
- 3 ÷ 3 = 1. Therefore, the prime factorization of 96 will be: 2 * 2 * 2 * 2 * 2 * 3
- Example 2: What will be the prime factorization of 120 using division method?
- Example 1: Prime factorization of 96 using division method will be to continuously divide 96 by a prime number it is no more divisible:
120 ÷ 2 = 60
60 ÷ 2 = 30
30 ÷ 2 = 15
15 ÷ 3 = 5
- 5 ÷ 5 = 1. Therefore, the prime factorization of 120 will be: 2 * 2 * 2 * 3 * 5
Another modern approach to solving prime factorization is to choose a random pair of factors and then split these factors until all of the factors are prime. This is just like the previous method above but the only difference is splitting the numbers. Let’s use one of the previous examples from above to determine prime factorization with splitting method and see if we will actually arrive at the same answer as before:
-
- Example 1: Prime factorization of 96 using division method will be to continuously divide 96 by a prime number it is no more divisible:
Step 1: Find two natural prime numbers that can give 96:
96
Step 2: Find the prime factorization of these two numbers:
8 12
Step 3: Repeat step 2
4 2 4 3
2222
-
- Therefore, the prime factorization of 90 is: 2 * 2 * 2 * 2 * 2 * 3
- Example 2: What will be the prime factorization of 120 using division method?
Step 1: Find two natural prime numbers that can give 120:
-
- 120
Step 2: Find the prime factorization of these two numbers:
10 12
Step 3: Repeat step 2
2 5 6 2
3 2
- Therefore, the prime factorization of 90 is: 2 * 2 * 2 * 3 * 5
In the example above, you can write out the prime factorization answer in any format that pleases you but it is preferable to write them out in the order of ascending i.e. starting from the smallest prime number to the highest
Other examples of prime factorization can be the below examples:
- Example 1: What will be the prime factorization of 12
- It is best to start from least prime number then proceed to the high one, in this case, we start with 2: 12 ÷ 2 = 6
- Notice that 2 can still divide through 6. So, we try again with 2: 6 ÷ 2 = 3
- Now we have 3 and it is not a prime number, which means we cannot divide any further
- Therefore, the prime factorization of 12 will be: 2 * 2 * 3
- Example 2: What is the prime factorization of 147?
- It is best to start from least prime number then proceed to the high one, in this case, we start with 2: 147 ÷ 2 = 73½. In this case, we cannot use 2 since only whole numbers are required as answers and not fractions. Let’s try the next smaller prime number, i.e. 3: 147 ÷ 3 = 49. Now we have a whole number which means we can proceed
- Notice that prime number 3, 4, 5, and 6 cannot divide through 49 without having a remainder. So, we try again with the next number after 6 which is 7: 49 ÷ 7 = 7. That worked
- Now, that is as far as it can go
- Therefore, the prime factorization of 147 will be: 3 * 7 * 7
- Example 3: What is the prime factorization of 17?
- This is kind of tricky if you think. First, 27 is already a prime number, which means it can only divide by 1 and itself alone. So that is as far as it can go i.e. 17 = 17
Why find prime factorization
We already known that any prime number can only be divided by 1 or by itself, meaning that it cannot be factored any further. While every other whole number can be broken down into prime number factors. In addition, prime numbers are the basic building blocks of all naturally occurring numbers. Prime factorization can come in handy while working with big numbers, such as in the case of cryptography
Positive integers without any prime factors in common are called coprime. Two integers a and b can also be defined as coprime if both integers have greatest common divisor that is equal to 1, i.e. gcd(a, b) = 1. Euclid’s algorithm can as well be used to find if two integers can be coprime without having to know their prime factors. The algorithm runs in a time that is polynomial in the number of digits that are involved. The integer 1 is coprime to all the positive integers, as well as itself. This is so because it has no prime factors as it is known to be an empty product. This means that gcd(1, b) = 1 for any b ≥ 1
Cryptography is known to be the study of codes. prime factorization is very important to people who try to make and also break secret codes using numbers. That is because factoring large numbers can be difficult sometimes, and can take computers a long time to do. In determining the prime factors for any set of giving number can be a perfect example of some problems that are frequently used to check cryptographic security in systems of encryption. This problem is also believed to require a super-polynomial time in the number of digits. Furthermore, it is relatively very easy to fabricate a problem that would take longer than the known age of the universe to solve on the set of computers we have in our today’s world using current algorithms.
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