Basic Algebra

Integer Factor Calculator

Find all the factors, complete factor pairs, and determine the prime or composite status of any number.

Preparing Factor Calculator

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Enter any whole integer (positive or negative) below to resolve the complete list of its positive and negative factors.

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A Guide to Understanding Factors and Divisibility

In basic mathematics and algebra, a factor is any whole number (integer) that divides into another number evenly, leaving absolutely no remainder. For example, because the number 12 can be divided by 3 exactly 4 times, both 3 and 4 are factors of 12. Factors are crucial in grouping physical objects, simplifying fraction equations, solving quadratic trinomials, and understanding basic cryptographical codes.

Factor Pairs & Prime Status

Factors always appear in pairs. A factor pair consists of two integers that, when multiplied together, produce the target number. For example, the factor pairs of 12 are (1, 12), (2, 6), and (3, 4). If a number is negative, one factor in each pair must be negative (e.g. -2 × 6 = -12).

If a positive integer has exactly two factors—namely, 1 and itself—it is classified as a prime number. If it has more than two factors, it is a composite number. To see only the prime building blocks of a number, use our dividing numbers into prime components. To find the largest shared factor between multiple numbers, check out our greatest common divisor solver.

Everyday Factor Scenarios

Arranging Spaces: An event organizer with 24 tables uses factor pairs to decide room layouts (e.g. 3 rows of 8, or 4 rows of 6 tables).
Simplifying Fractions: Dividing top and bottom numbers by common factors reduces fractions to lowest terms. You can evaluate fractions with our visual fractions solver.
Scheduling and Multiple Alignment: Factors help in estimating repeating cycles, which you can coordinate with our least common multiple solver.
Basic Computations: Resolving factors is a basic part of daily calculations, which you can double-check with our standard daily math tools.

Finding Factors Systematically

To find all factors of a number, you start testing division from 1 and move upwards. For example, to factor 20: 20 ÷ 1 = 20 (factors 1, 20); 20 ÷ 2 = 10 (factors 2, 10); 20 ÷ 3 leaves a remainder; 20 ÷ 4 = 5 (factors 4, 5).

Once you reach the square root of the number (which is around 4.47 for 20), you can stop testing because any larger factors will correspond to pairs you have already discovered. Our online tool handles this search instantly and lists the factors in order.

You can also use quick divisibility rules to speed up the process. For example, a number is divisible by 2 if its last digit is even, divisible by 3 if the sum of its digits is divisible by 3, and divisible by 5 if it ends in 0 or 5. These rules make factoring by hand much easier when verifying small datasets.

Example of Arranging Chairs

Suppose a teacher wants to arrange 30 student chairs in a classroom in equal rows.

By inputting 30 into the factor calculator, we find the factors are 1, 2, 3, 5, 6, 10, 15, and 30. The factor pairs are (1, 30), (2, 15), (3, 10), and (5, 6). The teacher can set up 2 rows of 15, 3 rows of 10, or 5 rows of 6 chairs. This simple example shows how factor pairs provide immediate solutions for layout planning.