Factor Calculator

About Factor Calculator

A factor is an integer that divides another integer exactly without leaving a remainder. Factors describe the multiplication structure of a number. For example, 28 can be written as 1 × 28, 2 × 14, or 4 × 7, so 1, 2, 4, 7, 14, and 28 are all positive factors of 28. This calculator finds the complete set of positive factors and explains the prime factorization of the number.

Factors always occur in pairs. If a × b = n, then both a and b are factors of n. For example, when calculating the factors of 36, 1 × 36, 2 × 18, 3 × 12, 4 × 9, and 6 × 6 are factor pairs. This relationship allows factors to be found efficiently by testing divisors only up to the square root of the number.

The calculator also determines the prime factorization. Prime factorization expresses a positive integer greater than 1 as a product of prime numbers. For example, 60 = 2 × 2 × 3 × 5, which can also be written as 2² × 3 × 5. The prime factorization of an integer greater than 1 is unique apart from the order of the prime factors, a result known as the Fundamental Theorem of Arithmetic.

What Are Factors?

Factors are numbers that divide a given number evenly. If the remainder after division is zero, the divisor is a factor.

For example, the factors of 12 are:

Each of these numbers divides 12 exactly. The corresponding factor pairs are 1 × 12, 2 × 6, and 3 × 4.

Factor Pairs

A factor pair consists of two integers whose product equals the original number. Finding factor pairs is one of the simplest ways to understand the complete factor structure of an integer.

Therefore, the positive factors of 28 are 1, 2, 4, 7, 14, and 28. The factors can be listed in ascending order for easier comparison.

Prime Factorization

A prime number has exactly two positive factors: 1 and itself. Examples include 2, 3, 5, 7, 11, 13, and 17. A composite number has more than two positive factors and can be expressed as a product of smaller prime numbers.

Prime factorization breaks a composite number down until only prime factors remain. For example:

Prime Numbers and Composite Numbers

A number greater than 1 is either prime or composite. A prime number cannot be expressed as a product of two smaller positive integers, while a composite number can.

The number 1 is a special case. It has exactly one positive factor, itself, and is therefore neither prime nor composite.

How the Calculator Finds Factors

It is not necessary to test every integer from 1 through n. If a number n has a factor greater than √n, it must have a corresponding factor smaller than √n. Therefore, testing possible divisors up to √n is sufficient to identify every factor pair.

For example, √28 is approximately 5.29. Testing 1, 2, 3, 4, and 5 is enough to discover the factor pairs of 28. The successful pairs are 1 × 28, 2 × 14, and 4 × 7.

Factors and Prime Factorization

Every positive factor of a number can be constructed from combinations of its prime factors. For example, the prime factorization of 60 is 2² × 3 × 5. By choosing different combinations of these prime factors, the positive factors of 60 can be generated.

Number of Factors

The prime factorization can also be used to determine how many positive factors a number has. If n = p₁a ^a × p₂b^b × ..., then the number of positive factors is found by adding 1 to each exponent and multiplying the resulting values.

For example:

Perfect Squares

A perfect square is an integer that can be written as the square of another integer. Examples include 1, 4, 9, 16, 25, 36, and 49. A useful property is that the prime-factor exponents of a perfect square are all even.

Perfect Cubes

A perfect cube is an integer that can be written as the cube of another integer. Examples include 1, 8, 27, 64, and 125. In the prime factorization of a perfect cube, every prime exponent is a multiple of 3.

Proper Factors

Proper factors are the positive factors of a number excluding the number itself. They are useful when classifying numbers according to the relationship between a number and the sum of its proper factors.

Perfect, Abundant, and Deficient Numbers

A positive integer can also be classified by comparing it with the sum of its proper factors. A perfect number has a proper-factor sum equal to the number itself. An abundant number has a larger sum, while a deficient number has a smaller sum.

The smallest perfect number is 6 because 1 + 2 + 3 = 6.

Why Prime Factorization Is Useful

Prime factors are important in many areas of mathematics. They can be used to simplify fractions, determine the greatest common factor (GCF), calculate the least common multiple (LCM), analyze divisibility, identify perfect powers, and solve problems involving number theory.

For example, prime factorization provides a systematic way to find the GCF of two numbers by identifying the prime factors they share. It can also be used to determine the LCM by taking the highest power of every prime appearing in either factorization.

Example: Factors of 28

How to Use the Factor Calculator

Enter a positive whole number in the Number field and select Calculate. The calculator identifies every positive factor, organizes them in ascending order, displays the factor pairs, and determines the prime factorization.

The result also provides additional information about the number, including whether it is prime or composite, the number of factors, the sum of its factors, proper-factor information, and whether it is a perfect square or perfect cube. The prime-factorization section shows the division steps used to break the number into prime factors.

Important Mathematical Notes

This calculator works with positive integers. Zero and negative integers require different definitions when discussing their complete sets of divisors, while decimal values are not included in the standard positive-integer factorization problem.

For very large integers, factorization can require substantially more computation than simple arithmetic. The calculator therefore uses integer arithmetic within the numerical range safely supported by the browser.