How to Find How Many Factors Are in a Number

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Last Updated: January 24, 2023 References

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

1
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.
2
Calculate the Prime Factorization of the number.^{[2]
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Research source} There are many methods of doing this, but usually, the simplest way is to make a factor tree.^{[3]
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Research source} 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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3
Take all the exponents and add one to each of them.^{[4]
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Research source}
- In the example 2³ and 3², the exponents are 3 and 2 - adding one to each will make them 4 and 3.
4
Multiply the modified exponents together.^{[5]
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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 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.
Question

How many factors does each number have?

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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.
Question

What is the fastest way to find the factors of a number?

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Divide 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.”

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.