Factor Calculator

A factor calculator helps you find the whole numbers that divide evenly into another number. Enter a positive integer to see its complete list of factors, factor pairs, prime factorization, number of factors, and other useful details. You can also compare two or more numbers to find common factors and the greatest common factor, or switch to the polynomial mode to factor a quadratic expression when it can be written as a product of integer binomials.

This page is built for more than one type of factoring problem. Use the standard factor calculator when you need the factors of a number such as 30, 20, 15, 10, 25, or 6. Use Prime Factorization to break a composite number into prime factors. Use Common Factors & GCF when you are comparing several integers, or use Factor a Polynomial for quadratic expressions in the form ax² + bx + c. The results are shown step by step so you can use the calculator to check your work instead of only seeing a final answer.

How to Use the Factor Calculator

The calculator has four modes because the word factoring can describe several related math problems. The first mode finds the positive factors of one whole number. The second breaks a number into prime factors. The third compares several numbers and finds the factors they have in common. The fourth is for factoring quadratic polynomials with integer coefficients.

Factors of a Number

Select Factors of a Number, enter a positive integer, and click Find Factors. The result includes every positive factor in order, the factor pairs that multiply to the original number, the prime factorization, the total number of positive factors, the sum of those factors, and whether the number is prime, composite, or neither.

This is the best mode for searches such as factors of 30, factors of 20, factors of 15, factors of 10, factors of 25, or factors of 6. You can change the input and calculate as many different integers as you need without reloading the page.

Prime Factorization

Select Prime Factorization when you want the number written only as a product of prime numbers. For example, 60 becomes 2² × 3 × 5. The calculator shows both the repeated prime factors and the shorter exponential form.

Common Factors and GCF

Choose Common Factors & GCF when you are comparing two or more numbers. Enter the integers separated by commas, such as 24, 36, 60. The calculator first finds their greatest common factor and then lists all factors of that GCF. Those are exactly the positive factors shared by every number in the set.

Factor a Polynomial

Select Factor a Polynomial when you have a quadratic in the form ax² + bx + c. Enter the three integer coefficients. If the expression factors into integer binomials, the calculator returns the factored form. For example, x² + 5x + 6 becomes (x + 2)(x + 3). If the quadratic does not factor over the integers, the result explains why instead of forcing an incorrect answer.

What Is a Factor?

A factor is a whole number that divides another whole number evenly, leaving no remainder. If a number n can be written as a × b, then a and b are factors of n. For example, 3 is a factor of 18 because 18 ÷ 3 = 6 with no remainder. Since 3 × 6 = 18, both 3 and 6 are factors of 18.

The word factor can be used as both a noun and a verb in mathematics. As a noun, it refers to one of the numbers being multiplied. As a verb, to factor a number or expression means to rewrite it as a product of simpler parts. That is why a factoring calculator can refer to integer factoring, prime factorization, or algebraic factoring depending on the problem.

Every positive integer has at least one positive factor: 1. Every positive integer greater than 1 also has itself as a factor. Prime numbers have exactly two positive factors, 1 and the number itself. Composite numbers have more than two positive factors. The number 1 is a special case because it has only one positive factor and is classified as neither prime nor composite.

Factors are foundational in arithmetic because they appear in fraction simplification, common denominators, divisibility, greatest common factors, least common multiples, ratios, algebra, and number theory. If you are simplifying a fraction after finding a common factor, the fraction calculator can reduce the result and handle other fraction operations.

How to Find the Factors of a Number

One way to find the factors of a number is to test every positive integer from 1 up to the number and keep the values that divide evenly. That method works, but it performs far more checks than necessary. A faster approach is to test possible divisors only up to the square root of the number. Whenever one divisor is found, a matching factor is found at the same time.

Take 36 as an example. Start with 1. Since 36 ÷ 1 = 36, the first factor pair is 1 and 36. Next, 36 ÷ 2 = 18, so 2 and 18 form another pair. Dividing by 3 gives 12. Dividing by 4 gives 9. The next whole-number divisor is 6, and 6 × 6 = 36. At that point you have reached the square root of 36, so every positive factor has been found.

This square-root method is efficient because factors come in pairs. If 2 divides 36, then 18 is automatically another factor. If 4 divides 36, then 9 is automatically another factor. The calculator uses this paired approach for the standard factor mode.

You can also check the multiplication or division behind a factor pair with the simple math calculator if you want to verify the arithmetic separately.

What Are Factor Pairs?

A factor pair is a pair of whole numbers whose product equals the original number. The factor pairs of 20 are 1 × 20, 2 × 10, and 4 × 5. Listing factors as pairs is often the easiest way to make sure none have been missed.

Perfect squares contain one pair in which both factors are the same. For 25, the factor pairs are 1 × 25 and 5 × 5. The repeated 5 is only included once in the ordinary list of positive factors, so the factors of 25 are 1, 5, and 25.

Factor pairs are also useful in algebra. When factoring a quadratic such as x² + 7x + 12, you look for two integers that multiply to 12 and add to 7. The pair 3 and 4 satisfies both conditions, giving (x + 3)(x + 4).

Prime Factorization

Prime factorization means rewriting a positive integer greater than 1 as a product of prime numbers. A prime number has exactly two positive factors: 1 and itself. Examples include 2, 3, 5, 7, 11, 13, and 17.

Consider 84. Since it is even, divide by 2 to get 42. Divide by 2 again to get 21. Then divide 21 by 3 to get 7. Since 7 is prime, the process stops. The prime factorization is 2 × 2 × 3 × 7, which can also be written as 2² × 3 × 7.

Prime factorization is useful because it exposes the basic building blocks of a composite number. It can be used to find a greatest common factor, least common multiple, simplify fractions, study divisibility, and work with algebraic expressions. Khan Academy also provides a useful factors, multiples, and prime factorization unit if you want additional practice beyond the calculator.

How a Factor Tree Works

A factor tree is a visual way to reach the same prime factorization. Start by splitting a composite number into any factor pair other than 1 and itself. Continue splitting composite branches until every branch ends in a prime number. Different valid trees may use different intermediate factors, but the final collection of prime factors will be the same.

For 72, you could start with 8 × 9. Then 8 becomes 2 × 4, 4 becomes 2 × 2, and 9 becomes 3 × 3. The result is 2 × 2 × 2 × 3 × 3, or 2³ × 3². You could instead begin with 6 × 12 and still arrive at the same prime factorization.

Common Factors and the Greatest Common Factor

A common factor is a factor shared by two or more numbers. The common factors of 18 and 30 are the values that divide both numbers evenly. The factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. Their common factors are 1, 2, 3, and 6.

The greatest common factor, often shortened to GCF, is the largest of those shared values. For 18 and 30, the GCF is 6. You may also see the terms greatest common divisor or GCD. In ordinary integer arithmetic, GCF and GCD refer to the same underlying idea.

The calculator finds the GCF first and then factors that number. This works because every common factor of the original numbers must also be a factor of their greatest common factor. If the GCF of several numbers is 12, for example, the common positive factors are the positive factors of 12: 1, 2, 3, 4, 6, and 12.

Finding a GCF by Prime Factorization

You can find a GCF manually by writing the prime factorization of each number and selecting only the prime factors they share. Use the lowest exponent that appears in every number.

For 48 and 72, the prime factorizations are 48 = 2⁴ × 3 and 72 = 2³ × 3². Both contain powers of 2 and 3. The smaller power of 2 is 2³, and the smaller power of 3 is 3¹. Multiplying 2³ × 3 gives 24, so the GCF is 24.

Factoring Polynomials

Factoring a polynomial means rewriting an algebraic expression as a product. The polynomial mode on this page focuses on quadratics of the form ax² + bx + c with integer coefficients. This includes common classroom problems such as x² + 5x + 6, 2x² + 5x + 2, and 6x² + 11x + 3.

For a monic quadratic, where a = 1, the basic approach is to look for two numbers that multiply to c and add to b. In x² + 5x + 6, the numbers 2 and 3 multiply to 6 and add to 5. Therefore the factored form is (x + 2)(x + 3).

When the leading coefficient is not 1, factoring can require more steps. For 2x² + 5x + 2, the factors are (2x + 1)(x + 2). Expanding the product gives 2x² + 4x + x + 2, which simplifies back to 2x² + 5x + 2.

The polynomial calculator checks the discriminant b² − 4ac. If the discriminant is a nonnegative perfect square, the roots are rational and the quadratic can be expressed using rational linear factors. For the integer-coefficient quadratics handled here, the calculator formats the factorization using integer binomials whenever possible.

If the three coefficients share a greatest common factor, that common factor should be taken out first. For 3x² + 12x + 9, every coefficient is divisible by 3. Factoring out 3 gives 3(x² + 4x + 3), and the quadratic inside becomes 3(x + 1)(x + 3).

What Does Factored Form Mean?

Factored form is an expression written as multiplication rather than expanded addition and subtraction. The expanded polynomial x² + 7x + 10 can be written in factored form as (x + 5)(x + 2). Both expressions represent the same polynomial. Factored form is especially useful when solving equations because each factor can be set equal to zero.

For example, if (x + 5)(x + 2) = 0, then either x + 5 = 0 or x + 2 = 0. The solutions are x = −5 and x = −2. Factoring connects the structure of a polynomial with its roots, which is one reason it is a major topic in algebra.

When a Quadratic Does Not Factor Over the Integers

Not every quadratic with integer coefficients can be factored into integer binomials. For x² + x + 1, the discriminant is 1 − 4 = −3. A negative discriminant means there are no real roots, so there is no factorization into real linear factors.

Another example is x² + x − 1. Its discriminant is 5, which is positive but not a perfect square. The roots are real but irrational, so the expression does not factor into integer binomials. In situations like these, a scientific calculator can be useful for evaluating square roots or checking decimal approximations.

Factor Examples

The examples below cover several of the specific factor searches students commonly make. Each list contains positive factors only, which is the standard convention for elementary factor problems unless negative factors are requested.

Factors of 30

The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The factor pairs are 1 × 30, 2 × 15, 3 × 10, and 5 × 6. Its prime factorization is 2 × 3 × 5. Because it has more than two positive factors, 30 is a composite number.

Factors of 20

The factors of 20 are 1, 2, 4, 5, 10, and 20. The factor pairs are 1 × 20, 2 × 10, and 4 × 5. Its prime factorization is 2² × 5. The repeated 2 in the prime factorization is written with an exponent of 2.

Factors of 15

The factors of 15 are 1, 3, 5, and 15. Its factor pairs are 1 × 15 and 3 × 5. The prime factorization is 3 × 5. Since 3 and 5 are both prime, the factorization is already complete.

Factors of 10

The factors of 10 are 1, 2, 5, and 10. Its factor pairs are 1 × 10 and 2 × 5, and the prime factorization is 2 × 5. Any whole number ending in 0 is divisible by both 2 and 5, which makes those two factors easy to identify.

Factors of 25

The factors of 25 are 1, 5, and 25. The factor pairs are 1 × 25 and 5 × 5. Since 25 is a perfect square, the final pair repeats the same number. Its prime factorization is 5².

Factors of 6

The factors of 6 are 1, 2, 3, and 6. The factor pairs are 1 × 6 and 2 × 3. The prime factorization is 2 × 3. This is one of the smallest composite numbers and is a useful early example for learning the difference between a prime number and a composite number.

How Many Factors Does a Number Have?

You can count the number of positive factors directly from a factor list, but prime factorization gives a faster formula for larger numbers. If the prime factorization of a number is pᵃ × qᵇ × rᶜ, then the number of positive factors is (a + 1)(b + 1)(c + 1).

For 72, the prime factorization is 2³ × 3². Add 1 to each exponent and multiply: (3 + 1)(2 + 1) = 4 × 3 = 12. Therefore 72 has 12 positive factors.

This works because a factor of 72 can contain 2 to the power 0, 1, 2, or 3, giving four choices, and 3 to the power 0, 1, or 2, giving three choices. Every combination produces one positive factor, for 4 × 3 = 12 possibilities.

Prime Numbers and Composite Numbers

A prime number is a whole number greater than 1 with exactly two positive factors. A composite number is a whole number greater than 1 with more than two positive factors. This distinction matters because prime numbers cannot be broken into smaller positive integer factors other than 1 and themselves, while composite numbers can.

For example, 13 is prime because its only positive factors are 1 and 13. The number 12 is composite because its factors are 1, 2, 3, 4, 6, and 12. The number 1 is neither prime nor composite because it has only one positive factor.

If you are doing a larger calculation involving powers, roots, or factorials after finding factors, the online calculator is another general-purpose tool available on 247Calculator.

Divisibility Rules That Make Factoring Faster

Divisibility rules help you decide whether a number has a particular factor without completing long division. They are especially useful when finding factors by hand or building a prime factorization.

• A number is divisible by 2 if its last digit is even.
• A number is divisible by 3 if the sum of its digits is divisible by 3.
• A number is divisible by 4 if its last two digits form a number divisible by 4.
• A number is divisible by 5 if it ends in 0 or 5.
• A number is divisible by 6 if it is divisible by both 2 and 3.
• A number is divisible by 8 if its last three digits form a number divisible by 8.
• A number is divisible by 9 if the sum of its digits is divisible by 9.
• A number is divisible by 10 if it ends in 0.

For example, consider 1,260. It ends in 0, so it is divisible by 2, 5, and 10. Its digits add to 9, so it is also divisible by 3 and 9. Since it is divisible by both 2 and 3, it is divisible by 6 as well. These quick checks can reveal several factors before you begin a full factor search.

Factors vs. Multiples

Factors and multiples are closely related but point in opposite directions. A factor divides a number evenly. A multiple is produced by multiplying a number by an integer. For the number 6, the positive factors are 1, 2, 3, and 6. Some positive multiples of 6 are 6, 12, 18, 24, 30, and 36.

A number has a finite number of positive factors but infinitely many positive multiples. This is why you can list every positive factor of 30, but you cannot list every positive multiple of 30.

The relationship can also be stated in reverse. If 4 is a factor of 20, then 20 is a multiple of 4. If 7 is a factor of 35, then 35 is a multiple of 7.

Factors and Fractions

Factoring is one of the main ideas used to simplify fractions. To reduce a fraction, divide its numerator and denominator by a common factor. Using the greatest common factor reduces the fraction to lowest terms in one step.

For 24/36, the GCF of 24 and 36 is 12. Dividing both values by 12 gives 2/3. You could also divide by smaller common factors in several steps, but using the GCF is faster. Our fraction calculator can simplify fractions automatically and also add, subtract, multiply, and divide them.

Factors and Least Common Multiples

Prime factors can also be used to find the least common multiple, or LCM. When finding a GCF from prime factorizations, you keep only shared prime factors using the smallest exponents. When finding an LCM, you include every required prime factor using the largest exponent that appears.

For 12 and 18, the prime factorizations are 12 = 2² × 3 and 18 = 2 × 3². Their GCF uses the smaller powers: 2 × 3 = 6. Their LCM uses the larger powers: 2² × 3² = 36.

Why Factoring Matters in Algebra

In arithmetic, factoring is mostly about divisibility and number structure. In algebra, the same idea becomes a method for rewriting expressions. You look for common factors or patterns that let an expression be written as multiplication.

One of the first algebraic techniques is taking out a greatest common factor. In 6x² + 9x, both terms contain 3x. Factoring out 3x gives 3x(2x + 3). This is the algebraic version of recognizing that several whole numbers share a common divisor.

Quadratic factoring goes a step further. An expression such as x² + 8x + 15 can be rewritten as (x + 3)(x + 5). Factored form can reveal where the polynomial equals zero and often makes equations easier to solve.

Common Factoring Mistakes

One common mistake is forgetting 1 and the number itself when listing factors. Another is listing multiples instead of factors. If you are finding factors of 12, values such as 24 and 36 are multiples, not factors, because they do not divide 12.

Another mistake is stopping a prime factorization too early. Writing 60 = 6 × 10 is a valid factorization, but it is not a prime factorization because 6 and 10 are both composite. Continue until every factor is prime: 60 = 2² × 3 × 5.

With polynomials, a common error is finding two numbers that multiply correctly but do not produce the required middle term. Always expand your factored answer to check it. For example, (x + 2)(x + 4) expands to x² + 6x + 8, not x² + 5x + 8.

Small arithmetic errors can also cause a correct factoring method to fail. If you need to check a percentage or another calculation while working through a longer problem, the percentage calculator is available for percentage-based math.

Using the Factor Calculator to Check Your Work

A calculator is most useful when it helps you understand where an answer came from. If you are learning factors, try solving the problem manually first. Write down the factor pairs you can find, then enter the number in the calculator and compare your list with the result.

For prime factorization, divide the number repeatedly by the smallest prime factor you can find. When you think you have reached the final prime product, check it with the Prime Factorization mode. For common factors, list the factors of each number separately before comparing them with the calculator’s result.

For polynomial factoring, expand the binomials after you get an answer. Multiplying the factors back together should recreate the original expression exactly. This reverse check is one of the quickest ways to catch a sign or coefficient mistake.

Factor Calculator Frequently Asked Questions

Final Thoughts

Factoring connects basic division with some of the most important ideas in arithmetic and algebra. At the simplest level, a factor is a number that divides another number evenly. From there, the same idea leads to factor pairs, prime factorization, common factors, greatest common factors, fraction simplification, and polynomial factoring.

Use the Factors of a Number mode when you want a complete factor list, Prime Factorization when you want the number broken into primes, Common Factors & GCF when several integers are involved, and Factor a Polynomial for quadratic expressions. The calculator is designed to show enough information to verify your work while keeping each result easy to read.

If you are using factoring as part of schoolwork and want to check another type of result, you can also use the grade calculator to track test scores, weighted assignments, finals, and GPA.