What is a factor?
In multiplication, factors are the integers that are multiplied together to find other integers. For example, 6 × 5 = 30. In this example, 6 and 5 are the factors of 30. 1, 2, 3, 10, 15, and 30 would also be factors of 30. Essentially, an integer a is a factor of another integer b, so long as b can be divided by a with no remainder. Factors are important when working with fractions, as well as when trying to find patterns within numbers.
Prime factorization involves finding the prime numbers that, when multiplied, return the number being addressed. For example, prime factorization of 120 results in 2 × 2 × 2 × 3 × 5. It can be helpful to use a factor tree when computing the prime factorizations of numbers. Using 120:
From the simple example of 120, it is clear that prime factorization can become quite tedious fairly quickly. Unfortunately, there is currently no known algorithm for prime factorization that is efficient for very large numbers. Many calculators, including the one on this page, cannot calculate prime factorizations beyond a certain magnitude. One concerted effort between several researchers to factor RSA-768, a 232-digit number, took 2 years using hundreds of machines.
Although no efficient algorithm has been found, it also has yet to be proven that no such algorithm exists, leaving room for anyone interested in having their name in a mathematical algorithm to formulate one (or prove in some self-named theorem that one doesn’t exist)!
To find the factors of a given number, you can follow these steps:
- Choose a positive integer to find its factors.
- Make a list of all the numbers that divide the given number without leaving any remainder.
- The numbers in the list are the factors of the given number.
For example, let’s say you want to find the factors of the number 24:
- Choose the number 24.
- Make a list of all the numbers that divide 24 without leaving a remainder: 1, 2, 3, 4, 6, 8, 12, and 24.
- Therefore, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
You can also use the prime factorization method to find the factors of a number. To do this, you need to factorize the number into its prime factors and then find all possible combinations of these prime factors.