Factor Calculator

What Is a Factor Calculator?

A Factor Calculator is a free online math tool that finds all positive factors of a whole number. Factors are numbers that divide another number exactly without leaving a remainder.

Our Factor Calculator also displays factor pairs, prime factorization, and the total number of factors. You can enter a second number to find common factors and the greatest common factor (GCF).

For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each divides 12 without a remainder.

Every positive integer greater than 1 has at least two positive factors: 1 and itself. The number 1 has only one positive factor.

How to Use the Factor Calculator

Follow these steps to find factors of one or two numbers:

1. Enter Number 1: Type the positive whole number whose factors you want to calculate.

2. Enter Number 2 (Optional): Add another positive whole number to compare factors and find the GCF.

3. Click Calculate: The calculator processes your numbers and displays the results.

4. Review the Results: Check the factors, factor pairs, prime factorization, and total factor count. If you entered two numbers, you will also see their common factors and GCF.

5. Click Clear: Reset the fields to perform another calculation.

You can leave Number 2 empty if you only want the factors of a single number.

Understanding Your Factor Calculator Results

The calculator provides several results depending on whether you enter one or two numbers.

For example, if you enter 48 and 10, the calculator identifies their shared factors as 1 and 2, with a GCF of 2.

If you only need to calculate the largest shared factor, you can also use our Greatest Common Factor Calculator.

How to Find Factors of a Number

To find factors manually, divide the number by positive whole numbers and check whether the remainder is zero.

You only need to test possible divisors up to the square root of the number because factors occur in pairs.

For a positive integer (n), test possible divisors (d) up to:

$d\leq\sqrt{n}$

When (d) divides (n) exactly, its paired factor is:

$\frac{n}{d}$

Example: Find the Factors of 36

Start by identifying pairs of numbers whose product is 36.

The positive factors of 36 are:

1, 2, 3, 4, 6, 9, 12, 18, 36

Notice that the factor pair (6, 6) contains the same number twice. Since 36 is a perfect square, 6 is counted only once in the list of distinct factors.

For more information about divisibility and factors, see Wolfram MathWorld's Divisor reference.

Factor Calculator Formula

The following mathematical formulas explain how factors, factor pairs, prime factorization, and factor counts are calculated.

1. Factor Formula

A positive integer (d) is a factor of (n) when dividing (n) by (d) leaves no remainder.

$n\bmod d=0$

For example:

$24\bmod6=0$

Therefore, 6 is a factor of 24.

2. Factor Pair Formula

Two positive integers form a factor pair when their product equals the original number.

$a\times b=n$

For example:

$4\times12=48$

Therefore, (4, 12) is a factor pair of 48.

3. Prime Factorization Formula

Prime factorization expresses a positive integer greater than 1 as a product of prime numbers.

$n=p_1^{a_1}\times p_2^{a_2}\times\cdots\times p_k^{a_k}$

Here, each ($p_i$) is a distinct prime number, and each ($a_i$) is its positive integer exponent.

For example:

$48=2^4\times3$

The prime factorization of 48 is 2 × 2 × 2 × 2 × 3, while its distinct prime factors are 2 and 3.

To explore this process further, use our Prime Factorization Calculator or read the Prime Factorization guide from Wolfram MathWorld.

4. Total Number of Factors Formula

If a positive integer has the prime factorization:

$n=p_1^{a_1}\times p_2^{a_2}\times\cdots\times p_k^{a_k}$

The total number of positive factors is:

$d(n)=(a_1+1)(a_2+1)\cdots(a_k+1)$

For example:

$48=2^4\times3^1$

$d(48)=(4+1)(1+1)=10$

Therefore, 48 has 10 positive factors.

For the number 1, the total factor count is 1.

Factor Calculation Example: 48 and 10

Suppose you enter:

• Number 1: 48

• Number 2: 10

The calculator displays the following results:

Common Factors: 1, 2

Greatest Common Factor: 2

$\operatorname{GCF}(48,10)=2$

The GCF is 2 because it is the largest positive integer that divides both 48 and 10 without a remainder.

For a detailed explanation of this concept, see Wolfram MathWorld's Greatest Common Divisor reference.

Types of Factors in Mathematics

Factors can be classified into different groups based on their mathematical properties.

A prime number has exactly two positive factors: 1 and itself.

For example, 13 is prime because its only positive factors are 1 and 13.

You can check whether a number is prime using our Prime Number Calculator.

Factors vs. Multiples

Factors and multiples are related, but they describe different mathematical relationships.

A factor divides a number exactly, while a multiple is obtained by multiplying a number by an integer.

A positive integer has a finite number of positive factors but infinitely many positive multiples.

Factors are useful when simplifying fractions and finding common divisors. For fraction calculations, you can use our Fraction Calculator.

Multiples are useful when finding common denominators and solving scheduling or repetition problems. To find the smallest common multiple of two or more numbers, use our Least Common Multiple Calculator.