How to Find How Many Factors Are in a Number

Reviewed by Joseph Meyer

Last Updated: October 28, 2024 Approved

Finding how many factors are in a number is as easy a 1 2 3 if you know how to do it.^{[1]
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Research source} 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.^{[2]
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Expert Source

Taylor Klein
Advanced Math Teacher

Expert Interview}
- Take 72, for example, but the number could be denoted by a variable.
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Calculate the Prime Factorization of the number.^{[3]
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Research source} 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.^{[4]
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Expert Source

Taylor Klein
Advanced Math Teacher

Expert Interview}
- 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.^{[5]
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- In the example 2³ and 3², the exponents are 3 and 2 - adding one to each will make them 4 and 3.
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Multiply the modified exponents together.^{[6]
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Research source}
- 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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How do you factor a number?

Taylor Klein
Advanced Math Teacher
Taylor 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.

Taylor Klein

Advanced Math Teacher

Expert Answer

There 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.

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How many factors does each number have?

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This answer was written by one of our trained team of researchers who validated it for accuracy and comprehensiveness.

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This 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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How do you find all the factors of a number on a calculator?

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If you have a graphing calculator, you can do this by entering the equation y = [your number]/x and then finding all the whole number pairs of x’s and y’s in the resulting table or graph. You can also do a search for “factor calculator” online if you don’t have access to a graphing calculator.

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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).^{[7]
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This article is to tell you how many factors are in a number. This does not tell you how to Factor a Number.

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