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This article was reviewed by Joseph Meyer. Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University.
This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources.
This article has been viewed 365,314 times.
Finding how many factors are in a number is as easy a 1 2 3 if you know how to do it.[1] But for larger numbers, you can't just count one by one. This is a nice trick to find how many factors are in an integer.
Steps
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Identify the number. Any number in the world, but it's best to start with the simpler ones.- Take 72, for example, but the number could be denoted by a variable.
Calculate the Prime Factorization of the number.[2] There are many methods of doing this, but usually, the simplest way is to make a factor tree. This works because according to number theory, every integer (except -1, 0, and 1) has a number of prime numbers that, when multiplied together, will equal the number. Remember that 0 and 1 are not prime numbers.- 72 should be factorized into 2 and 36, 2, 6, and 6, and finally, 2, 2, 3, 2, 3, which equals 23*32.
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Take all the exponents and add one to each of them.[3]- In the example 23 and 32, the exponents are 3 and 2 - adding one to each will make them 4 and 3.
Multiply the modified exponents together.[4]- 4 x 3 = 12. There are 12 factors for the number 72 - 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.
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Expert Q&A
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- QuestionHow do you factor a number?
Taylor KleinTaylor Klein is an Advanced Math Teacher based in Philadelphia, Pennsylvania. She has worked in the education field for over 10 years, including eight years as a middle school Advanced Math Teacher. She has a master’s degree in Instructional Technology and Design and a master’s degree in Educational Leadership and Administration.
Advanced Math TeacherThere are several methods for factoring a number. One approach is to identify its factor pairs, such as with the number 20, where you find pairs like 1 times 20, 2 times 10, and 4 times 5. It's important to check each prime number initially and stop when you encounter repeated factors. Another method involves prime factorization, where you break the number into its prime factors. Taking 20 as an example again, you can break it down into 4 times 5. Since 5 is prime, it is circled and left untouched, while 4 is further broken down into the prime factors 2 and 2. Once all numbers are expressed as their prime factors, the factoring process is complete. - QuestionWhat is the fastest way to find the factors of a number?
How.com.vn Staff EditorThis answer was written by one of our trained team of researchers who validated it for accuracy and comprehensiveness.
Staff AnswerHow.com.vn Staff EditorStaff AnswerDivide the number by the smallest prime number above 1 that will fit into it and give you a whole number result (with no remainder). Do the same with the resulting number, and keep doing it until you reach the answer “1.” - QuestionHow many factors does each number have?How.com.vn Staff EditorThis answer was written by one of our trained team of researchers who validated it for accuracy and comprehensiveness.
Staff AnswerHow.com.vn Staff EditorStaff AnswerThis is a difficult question to answer, since there is an infinite number of numbers. Every number has at least 2 factors (1, and the number itself). Some numbers, known as “highly composite numbers,” can have very large numbers of factors. For example, 840 has 32 factors. For more information, check out the work of Achim Flammenkamp, who has created an algorithm for finding highly composite numbers.
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Examples
7540
- Prime Factorization - 225(29)(13). Now, because x1 = x, 29, 13, and 5 all have exponents to the first power.
- Add exponents by one. 3, 2, 2, 2.
- Multiply the modified exponents together. There are 24 factors of the number 7540.
15802
- Prime Factorization - 2(7901)
- Modify exponents - 2, 2
- Multiply. There are four factors of the number 15802 - 1, 2, 7901, 15802. 7901 is a prime number.
Tips
- The reason you add one to each exponent is the possibility of a number to the zeroth power. This means that, for the number 23, there are four combinations for the exponent which can be multiplied by another number: 20, 21, 22, and 23. You can multiply 20 by 72 which would still give you 72, as x0 = 1 (with the notable exception of 00, an indeterminate form).[5]Thanks
- This article is to tell you how many factors are in a number. This does not tell you how to Factor a Number.Thanks
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Expert Interview
Thanks for reading our article! If you’d like to learn more about mathematics, check out our in-depth interview with Taylor Klein.
References
- ↑ https://www.bbc.co.uk/bitesize/topics/zfq7hyc/articles/zp6wfcw
- ↑ https://www.mathsisfun.com/prime-factorization.html
- ↑ https://www.khanacademy.org/math/pre-algebra/pre-algebra-factors-multiples/pre-algebra-prime-factorization-prealg/v/prime-factorization
- ↑ https://www.mathsisfun.com/numbers/factors-all-tool.html
- ↑ https://www.cuemath.com/numbers/factors/
About This Article
Math Teacher
This article was reviewed by Joseph Meyer. Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University. This article has been viewed 365,314 times.
Co-authors: 29
Updated: April 4, 2024
Views: 365,314
Categories: Mathematics
Thanks to all authors for creating a page that has been read 365,314 times.
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