The Factoring Calculator finds the factors and factor pairs of a positive or negative number. Enter an integer number to find its factors.
For positive integers the calculator will only present the positive factors because that is the normally accepted answer. For example, you get 2 and 3 as a factor pair of 6. If you also need negative factors you will need to duplicate the answer yourself and repeat all of the factors as negatives such as -2 and -3 as another factor pair of 6. On the other hand this calculator will give you negative factors for negative integers. For example, -2 and 3 AND 2 and -3 are both factor pairs of -6.
Factors are whole numbers that are multiplied together to produce another number. The original numbers are factors of the product number. If a x b = c then a and b are factors of c.
Say you wanted to find the factors of 16. You would find all pairs of numbers that when multiplied together resulted in 16. We know 2 and 8 are factors of 16 because 2 x 8 = 16. 4 is a factor of 16 because 4 x 4 = 16. Also 1 and 16 are factors of 16 because 1 x 16 = 16. The factors of 16 are 1, 2, 4, 8, 16.
You can also think about factors in terms of division: The factors of a number include all numbers that divide evenly into that number with no remainder. Consider the number 10. Since 10 is evenly divisible by 2 and 5, you can conclude that both 2 and 5 are factors of 10.
The table below lists the factors for 3, 18, 36 and 48. It is important to note that every integer number has at least two factors: 1 and the number itself. If a number has only two factors that number is a prime number.
How to Factor Numbers: Factorization
This factors calculator factors numbers by trial division. Follow these steps to use trial division to find the factors of a number.
- Find the square root of the integer number n and round down to the closest whole number. Let's call this number s.
- Start with the number 1 and find the corresponding factor pair: n ÷ 1 = n. So 1 and n are a factor pair because division results in a whole number with zero remainder.
- Do the same with the number 2 and proceed testing all integers (n ÷ 2, n ÷ 3, n ÷ 4... n ÷ s) up through the square root rounded to s. Record the factor pairs where division results in whole integer numbers with zero remainders.
- When you reach n ÷ s and you have recorded all factor pairs you have successfully factored the number n.
Example Factorization Using Trial Division
Factors of 18:
- The square root of 18 is 4.2426, rounded down to the closest whole number is 4
- Testing the integer values 1 through 4 for division into 18 with a 0 remainder we get these factor pairs: (1 and 18), (2 and 9), (3 and 6). The factors of 18 are 1, 2, 3, 6, 9, 18.
Factors of Negative Numbers
All of the above information and methods generally apply to factoring negative numbers. Just be sure to follow the rules of multiplying and dividing negative numbers to find all factors of negative numbers. For example, the factors of -6 are (1, -6), (-1, 6), (2, -3), (-2, 3). See the Math Equation Solver Calculator and the section on Rules for Multiplication Operations.
Related Factoring Calculators
See our Common Factors Calculator to find all factors of a set of numbers and learn which are the common factors.
The Greatest Common Factor Calculator finds the greatest common factor (GCF) or greatest common divisor (GCD) of a set of numbers.
See the Least Common Denominator Calculator to find the lowest common denominator for fractions, integers and mixed numbers.