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Have you ever wondered how to break down a number into its prime factors? Parsing a number into its prime factorization allows us to express a given number as a product of its prime factors, providing insight into its fundamental building blocks. By understanding how to parse numbers into their prime factors, we can simplify complex mathematical calculations, solve problems related to prime numbers, and gain a deeper understanding of the relationships between numbers. In this guide, we will explore the concept of parsing a number into its prime factors, discussing various methods and strategies to accomplish this task effectively. So, if you’re ready to delve into the fascinating world of prime factorization, let’s get started!
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The factors of a given number are numbers that, when multiplied together, have the same product as the previous given number. Think of it another way, every number is a product of many factors. Learning to factorize – or break a number into factors – is an important math skill that applies not only in basic arithmetic, but also in algebra, calculus and more. See Step 1 to start learning how to factor a number!
Steps
Factoring Basic Integers
- Please choose number 12 . Write this number down on scratch paper.
- For our example, 12 has several factors like 12 × 1, 6 × 2, and 3 × 4 all equal 12. So we can say that the factor of 12 is 1, 2, 3, 4 , 6, and 12 . Let’s use the factors of 6 and 2 for the purpose of the lesson.
- Even numbers are especially easy to parse because every even number has a factor of 2. 4 = 2 × 2, 26 = 13 × 2, etc.
- According to our example, the number 12 has been decomposed into 2 × 6. Notice that 6 also has its own factor – 3 × 2 = 6. So we can say that 12 = 2 × (3 × 2) .
- In our example, 12 has been parsed as 2 × (2 × 3). 2, 2, and 3 are all prime numbers. If further analysis, we have to parse as (2 × 1) × ((2 × 1)(3 × 1)), which usually has no effect and is ignored.
- For example, let’s analyze -60. Accordingly:
- -60 = -10 × 6
- -60 = (-5 × 2) × 6
- -60 = (-5 × 2) × (3 × 2)
- -60 = -5 × 2 × 3 × 2 . Note that as long as the number of negative factors is odd, the product of all the factors will also be negative, similar to when there is only one negative factor. For example, -5 × 2 × -3 × -2 is also -60.
How to Factor Large Numbers
- For our example, let’s choose a 4-digit number to factor, which is 6.552 .
- In our example, since 6.552 is an even number, we know that 2 is the smallest prime factor of this number. 6.552 ÷ 2 = 3.276. In the left column, we write 2 , and 3.276 in the right column.
- Let’s continue the analysis. 3.276 ÷ 2 = 1.638, so we’ll write an extra 2 at the bottom of the left column, and 1.638 at the bottom of the right column. 1.638 ÷ 2 = 819, so we will write 2 and 819 at the bottom of the two columns as before.
- For our example, we get 819. 819 is odd, so 2 is not a factor of 819. Instead of writing 2, we’ll try the next prime number: 3. 819 ÷ 3 = 273 and there is no remainder, so we write 3 and 273 .
- When guessing factors, you should try all primes whose value is less than or equal to the square root of the largest factor you have found. If your number isn’t divisible by any of the factors, you’re probably trying to factor a prime, and the factoring process could end there.
- Let’s finish our numerical analysis. See detailed explanation below:
- Divide by 3: 273 ÷ 3 = 91, no remainder, so we write 3 and 91 .
- Let’s go on to the number 3: 3 is not a factor of 91, and the next smallest prime number (5) is also not a factor of 91, however 91 ÷ 7 = 13, there is no remainder, so we write 7 and 13 .
- Let’s keep trying with 7: 7 is not a factor of 13, so is 11 (the next prime number), but 13 has its own factor: 13 ÷ 13 = 1. So to complete the table analysis, we write 13 and 1 . We can stop the analysis here.
- In our example, 6.552 = 2 3 × 3 2 × 7 × 13 . This is the complete result after factoring 6,552 into primes. Regardless of the order in which the multiplication is performed, the final product will be 6.552.
Advice
- An important point is the concept of prime numbers: a number that has only two factors, 1 and itself. 3 is prime because its factors are only 1 and 3. Otherwise, 4 has another factor of 2. A number that is not prime is called a composite number . (The number 1 itself is neither prime nor composite — that’s a special case.)
- The smallest prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, and 23.
- Understand that a number is considered a factor of a larger number if the larger number is “divisible by the smaller” – that is, the larger number is divisible by the smaller number and leaves no remainder. For example, 6 is a factor of 24, because 24 ÷ 6 = 4 and there is no remainder. Conversely, 6 is not a factor of 25.
- Some numbers can be parsed in a faster way, but the above approach always works, plus, the prime factors are listed in ascending order as you’re done.
- Remember we are only talking about “natural numbers” – sometimes called “counts”: 1, 2, 3, 4, 5… We won’t go into negative numbers or fractions, That can be covered in separate articles.
- If the sum of the digits of a number is divisible by three, then three is a factor of that divisor. (819 has the sum of the digits 8+1+9=18, 1+8=9. Three is a factor of nine so it is also a factor of 819.)
Warning
- Don’t do unnecessary things. Once you’ve eliminated a factor value, you don’t need to try again. Once we are sure that 2 is not a factor of 819, we don’t need to retry with 2 for the rest of the process.
Things you need
- Paper
- Pens write, it is recommended to use pencils and erasers
- Computer (optional)
wikiHow is a “wiki” site, which means that many of the articles here are written by multiple authors. To create this article, 55 people, some of whom are anonymous, have edited and improved the article over time.
This article has been viewed 25,093 times.
The factors of a given number are numbers that, when multiplied together, have the same product as the previous given number. Think of it another way, every number is a product of many factors. Learning to factorize – or break a number into factors – is an important math skill that applies not only in basic arithmetic, but also in algebra, calculus and more. See Step 1 to start learning how to factor a number!
In conclusion, parsing a number into its prime factors can be approached using various techniques such as trial division, sieving, or prime factorization algorithms. Each method has its advantages and limitations depending on the size and complexity of the number. By understanding the fundamentals of prime numbers and implementing appropriate algorithms, one can efficiently parse a number into its prime factors. This process can be useful in various mathematical and computational applications, including cryptography, number theory, and data analysis. Overall, the ability to parse numbers into their prime factors not only aids in problem-solving but also deepens our understanding of the properties and structure of numbers.
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