Find all factors, factor pairs, and the prime factorization of any integer. See whether the number is prime or composite and follow the factor tree step by step.
Enter an integer and click Calculate to see its factors, factor pairs, and prime factorization.
Positive and negative whole numbers are accepted, including 0 and 1. This calculator computes exact factors for integers up to plus or minus 1,000,000,000,000 (one trillion).
A factor is an integer that divides another integer evenly, leaving no remainder. For example, consider 12. You can divide 12 ÷ 1 = 12, 12 ÷ 2 = 6, 12 ÷ 3 = 4, 12 ÷ 4 = 3, 12 ÷ 6 = 2, and 12 ÷ 12 = 1. Each division produces a whole-number result. Therefore the positive factors of 12 are 1, 2, 3, 4, 6, and 12.
Another way to recognize factors is through multiplication. 1 × 12 = 12. 2 × 6 = 12. 3 × 4 = 12. The numbers in each multiplication sentence form a factor pair. A number's factors tell us which integers can combine through multiplication to create that number.
A reliable way to find factors is to test smaller integers and look for divisions with no remainder. Suppose you want the factors of 36. Start with 1. 36 ÷ 1 = 36, so 1 and 36 form a factor pair. Try 2. 36 ÷ 2 = 18, so 2 and 18 form another pair. Try 3. 36 ÷ 3 = 12. Pair: 3 and 12. Try 4. 36 ÷ 4 = 9. Pair: 4 and 9. Try 5. 36 ÷ 5 does not produce a whole number, so 5 is not a factor of 36. Try 6. 36 ÷ 6 = 6. Pair: 6 and 6. Now you can stop. The positive factors are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
Factor pairs have an important property. For 36, the square root is 6. The factor pairs are 1×36, 2×18, 3×12, 4×9, and 6×6. Once you reach 6, all larger factors have already appeared as partners of smaller factors. This works because if both numbers in a factor pair were greater than the square root of the original number, their product would be greater than the original number. So when searching for factors of n, you only need to test possible divisors up to √n. That is much more efficient than testing every number from 1 all the way to n. If you need to calculate a square root separately, use the Square Root Calculator.
Factor pairs are two integers whose product equals the original number. For 24, the positive factor pairs are 1×24, 2×12, 3×8, and 4×6. Therefore the positive factors are 1, 2, 3, 4, 6, 8, 12, and 24. Factor pairs make it easier to find every factor without accidentally missing one.
Prime factorization expresses an integer greater than 1 as a product of prime numbers. A prime number has exactly two positive factors: 1 and itself. Examples of prime numbers include 2, 3, 5, 7, 11, 13, 17, and 19. Consider 60. You can write 60 = 6 × 10. Then factor those numbers further: 6 = 2 × 3, and 10 = 2 × 5. Therefore 60 = 2 × 3 × 2 × 5. Rearrange: 2 × 2 × 3 × 5. Using exponents: 2² × 3 × 5. That is the prime factorization of 60.
A factor tree repeatedly splits a composite number into smaller factors until every branch ends in prime numbers. Consider 360. One possible first split is 360 = 36 × 10. Then 36 = 6 × 6, and 10 = 2 × 5. Each 6 = 2 × 3. The prime leaves are 2, 2, 2, 3, 3, 5. Therefore 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5.
Yes. You might start 60 = 6 × 10. Someone else might start 60 = 3 × 20. Another person might use 60 = 4 × 15. The branches look different at first, but if each composite factor is broken down until only primes remain, the final prime factorization is the same except for the order of the factors. This is an important idea in number theory.
A prime number greater than 1 has exactly two positive factors: 1 and itself. Example: 29. 29 ÷ 1 = 29. 29 ÷ 29 = 1. There are no other positive integer divisors. Therefore 29 is prime.
A composite number greater than 1 has more than two positive factors. Example: 30. Factors: 1, 2, 3, 5, 6, 10, 15, 30. Because 30 has more than two positive factors, 30 is composite.
No. The number 1 has only one positive factor: 1. A prime number must have exactly two positive factors. Therefore 1 is not prime. It is also not composite. The number 1 is neither prime nor composite.
Zero behaves differently from ordinary positive integers. Every nonzero integer divides 0 evenly. For example, 0 ÷ 2 = 0. 0 ÷ 5 = 0. 0 ÷ 100 = 0. Because infinitely many nonzero integers divide zero, zero does not have a finite standard factor list like 12 or 36. Prime factorization of 0 is not defined. This calculator explains that instead of attempting to produce an endless factor list.
Yes. Consider -12. Positive factor magnitudes include 1, 2, 3, 4, 6, 12. The integer divisors include -12, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 12. The prime factorization can be written -1×2²×3. In many elementary factor problems, only positive factors are requested. That is why this calculator makes positive factors easy to see first and treats negative factors as additional information.
Prime factorization gives us a fast way to count positive factors. Suppose 360 = 2³×3²×5¹. Any factor of 360 can contain 2 zero, one, two, or three times. That gives 4 choices. It can contain 3 zero, one, or two times. That gives 3 choices. It can contain 5 zero or one time. That gives 2 choices. Multiply the number of choices: 4×3×2 = 24. Therefore 360 has 24 positive factors.
Proper factors are positive factors of a number excluding the number itself. Example: 12. All positive factors: 1, 2, 3, 4, 6, 12. Proper factors: 1, 2, 3, 4, 6. The term is useful in topics such as perfect numbers and divisor sums.
Prime factors are factors that are themselves prime numbers. Example: 12. All factors: 1, 2, 3, 4, 6, 12. Prime factors: 2 and 3. Prime factorization: 2²×3. Not every factor is a prime factor.
Factors divide a number evenly. Multiples are produced by multiplying a number. For 6, the positive factors are 1, 2, 3, 6. The positive multiples begin 6, 12, 18, 24, 30, 36, and continue. The factor list of a nonzero integer is finite. Its positive multiple list continues indefinitely.
A common factor divides two or more numbers evenly. Consider 12 and 18. Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. The values appearing in both lists are 1, 2, 3, 6. These are the common factors. The greatest of them is 6. So 6 is the greatest common factor of 12 and 18. CalculateThisWay does not currently have a dedicated Greatest Common Factor calculator, so this page explains common factors directly rather than linking to one.
The greatest common factor, often shortened to GCF, is the largest factor shared by two or more integers. Example: Factors of 20: 1, 2, 4, 5, 10, 20. Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30. Common factors: 1, 2, 5, 10. GCF: 10. The GCF is especially useful when simplifying fractions, simplifying ratios, and factoring algebraic expressions. Prime factorization can also be used to find the least common multiple, or LCM, of two numbers, although that is a separate calculation from the GCF.
Consider 18/24. Both 18 and 24 are divisible by 6. Divide both by 6. 18÷6=3. 24÷6=4. So 18/24 simplifies to 3/4. If you are using factors to simplify a fraction, the Fraction Calculator can reduce the fraction and show the arithmetic step by step.
Consider 18:24. Both values share a greatest common factor of 6. Divide both sides by 6. 18÷6=3. 24÷6=4. The simplified ratio is 3:4. Use the Ratio Calculator for ratio simplification and proportional calculations.
Prime factorization often contains repeated prime factors. Instead of writing 2×2×2×3×3×5, we can write 2³×3²×5. Exponent notation makes repeated multiplication easier to read. Prime factorization is often easier to read in exponent notation. If you need help evaluating powers such as 2³ or 3², use the Exponent Calculator.
Perfect squares contain a repeated center factor pair. Example: 144. Factor pair: 12×12. Therefore √144 = 12. The Square Root Calculator can calculate square roots, cube roots, and other roots.
Some small factors can be identified quickly using divisibility rules.
Consider 360. It ends in 0, so it is divisible by 2, 5, and 10. Its digit sum is 3+6+0 = 9. Therefore it is also divisible by 3 and 9.
No. The word factor can have different meanings depending on the math topic. This calculator focuses on integer factors. An integer factor divides another integer evenly. Example: 6 is a factor of 24 because 24÷6=4. A scale factor is used in geometry to enlarge or reduce similar figures. For example, a scale factor of 2 doubles corresponding lengths. CalculateThisWay does not currently have a dedicated scale-factor calculator, so no link is provided here; the two concepts should not be confused.
No. This calculator finds integer factors, factor pairs, and prime factorizations. Polynomial factoring is an algebra topic. Example: x²+5x+6 can be factored as (x+2)(x+3). That is different from listing the factors of an integer such as 36. CalculateThisWay does not currently have a dedicated polynomial-factoring tool, so none is linked here. This site's Quadratic Formula Calculator solves quadratic equations numerically, which is a related but different task from factoring a polynomial into binomials.
Once you move from finding factors of whole numbers into factoring algebraic expressions, the process becomes broader. For a visual walkthrough of several algebraic factoring methods, Mario's Math Tutoring provides a Factoring All Types Complete Guide video. This is a supplementary external learning resource, and it does not mean this integer Factor Calculator performs every type of factoring shown in the video.
Watch: Factoring All Types Complete GuideUse division. Suppose you want to know whether 8 is a factor of 56. Calculate 56÷8=7. The result is an integer with no remainder. Therefore 8 is a factor of 56. Now test 8 against 58. 58÷8=7.25. This is not an integer. Therefore 8 is not a factor of 58.
A number greater than 1 is prime if it has no positive divisors other than 1 and itself. You do not need to test every smaller number. As with factor searching, it is enough to test possible prime divisors up to the square root of the number. If none divide evenly, the number is prime. For broader arithmetic beyond factoring, such as everyday calculations alongside this one, the Scientific Calculator is useful, though it is not a substitute for prime factorization.
Every positive integer has 1 as a positive factor.
Every positive integer divides itself.
Factors usually come in pairs. If 3 divides 24, then 8 is the matching factor.
A factor must produce no remainder.
Factors divide into a number. Multiples are generated from a number.
Prime factors are only the factors that are prime.
1 is neither prime nor composite.
0 is neither prime nor composite.
Zero is divisible by every nonzero integer.
They are different mathematical concepts.
They require different methods.
Do not leave knowing only "the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36." Leave knowing the following:
And most importantly: I can see why each number in the factor list is there. That is what the CalculateThisWay Factor Calculator should help someone understand.
| Concept | Rule |
|---|---|
| Factor | Divides the number with no remainder |
| Factor Pair | a × b = n |
| Prime Number | Exactly 2 positive factors |
| Composite Number | More than 2 positive factors |
| Prime Factorization | Product of prime numbers |
| Range of Factor Search | Test candidates through √n |
| Number of Factors | If n = p^a × q^b × ..., count = (a+1)(b+1)... |
| GCF | Largest common factor shared by numbers |
| Proper Factor | Positive factor excluding the number itself |
| Perfect Square | Has a factor pair a × a |
Example 1: Number 12
Factors: 1, 2, 3, 4, 6, 12
Prime Factorization: 2² × 3
Status: Composite
Example 2: Number 29
Factors: 1, 29
Prime Factorization: 29
Status: Prime
Example 3: Number 36
Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factor pairs: 1×36, 2×18, 3×12, 4×9, 6×6
Prime Factorization: 2² × 3²
Perfect Square: Yes
Example 4: Number 60
Factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Prime Factorization: 2² × 3 × 5
Example 5: Number 360
Positive factors: 24 total
Prime Factorization: 2³ × 3² × 5
Example 6: Number 1
Factors: 1
Status: Neither prime nor composite
Example 7: Number 0
Standard factor list: Not applicable
Prime Factorization: Undefined
Example 8: Number -12
Positive factor magnitudes: 1, 2, 3, 4, 6, 12
Integer factors: ±1, ±2, ±3, ±4, ±6, ±12
Prime Factorization: -1 × 2² × 3
Find all positive factors of 28. Enter them separated by commas, smallest to largest.
Hint: test divisors starting at 1 and stop after reaching √28.
I have one integer and need its factors, factor pairs, or prime factorization.
You are hereI need to simplify, add, subtract, multiply, divide, or convert fractions.
Open Calculator ›What is a factor?
A factor is an integer that divides another integer evenly, leaving no remainder. For example, 3 is a factor of 12 because 12 ÷ 3 = 4 with nothing left over.
What is a factor in math?
In math, a factor is one of two or more integers that multiply together to produce a given number. Since 2 × 6 = 12, both 2 and 6 are factors of 12.
How do you find factors?
Test whole numbers starting at 1 and check whether each one divides the number evenly. Every successful divisor gives you a matching factor pair, and you only need to test up to the square root of the number.
How do you factor a number?
To factor a number, list every positive integer that divides it with no remainder, then pair each small factor with its larger partner. For 24, 1 pairs with 24, 2 pairs with 12, 3 pairs with 8, and 4 pairs with 6.
What is a factor calculator?
A factor calculator takes one integer and returns all of its positive factors, its factor pairs, its prime factorization, and whether it is prime, composite, or neither.
What are the factors of a number?
The factors of a number are every positive integer that divides it evenly. For 36, the factors are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
What are factor pairs?
Factor pairs are two integers that multiply together to equal the original number. For 24, the factor pairs are 1×24, 2×12, 3×8, and 4×6.
Why do factors come in pairs?
Every factor has a partner because dividing the number by that factor produces another integer, and multiplying the two back together returns the original number. The only exception is a perfect square, where the middle pair repeats the same number twice.
How do you know if a number is a factor?
Divide the larger number by the candidate number. If the division produces a whole number with no remainder, the candidate is a factor. 56 ÷ 8 = 7 exactly, so 8 is a factor of 56.
What is prime factorization?
Prime factorization expresses an integer greater than 1 as a product of prime numbers. The prime factorization of 360 is 2³ × 3² × 5.
How do you find prime factors?
Divide repeatedly by the smallest prime that divides evenly until only 1 remains, recording each prime used. For 60, that process gives 2, 2, 3, and 5.
What is a factor tree?
A factor tree is a diagram that repeatedly splits a composite number into smaller factors until every branch ends in a prime number.
Can factor trees be different?
Yes. You can start a factor tree with a different first split, but every complete factor tree for the same number ends with the same prime factorization, aside from the order the primes are written in.
What is a prime number?
A prime number greater than 1 has exactly two positive factors: 1 and itself. Examples include 2, 3, 5, 7, 11, and 13.
What is a composite number?
A composite number greater than 1 has more than two positive factors. 30 is composite because its factors are 1, 2, 3, 5, 6, 10, 15, and 30.
Is 1 prime?
No. The number 1 has only one positive factor, itself, and a prime number must have exactly two positive factors.
Is 1 composite?
No. A composite number must have more than two positive factors, and 1 has only one. The number 1 is neither prime nor composite.
Is 0 prime?
No. Zero does not have a finite list of positive factors, so it cannot be classified as prime.
Is 0 composite?
No. For the same reason, zero is not composite either. It is neither prime nor composite.
What are the factors of 0?
Zero does not have a standard finite factor list because every nonzero integer divides 0 evenly (0 ÷ n = 0 for any nonzero n). Prime factorization is not defined for 0.
Can negative numbers have factors?
Yes. Both positive and negative integers can divide a negative integer evenly. -12 has positive factor magnitudes of 1, 2, 3, 4, 6, and 12, and integer factors of -12, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, and 12.
What are proper factors?
Proper factors are all of a number's positive factors except the number itself. The proper factors of 12 are 1, 2, 3, 4, and 6.
What are prime factors?
Prime factors are the factors of a number that are themselves prime. The factors of 12 are 1, 2, 3, 4, 6, and 12, but only 2 and 3 are prime factors.
What is the difference between factors and prime factors?
All factors include every integer that divides the number evenly, while prime factors are only the factors that are also prime numbers. Prime factors are the building blocks used in the full prime factorization.
What is the difference between factors and multiples?
Factors divide into a number evenly and are finite in count. Multiples are created by multiplying a number and continue indefinitely. The factors of 6 are 1, 2, 3, and 6, while the multiples of 6 begin 6, 12, 18, 24, and continue forever.
What is a common factor?
A common factor is a number that divides two or more integers evenly. 12 and 18 share the common factors 1, 2, 3, and 6.
What is the greatest common factor?
The greatest common factor, or GCF, is the largest factor shared by two or more integers. The GCF of 12 and 18 is 6.
How do factors help simplify fractions?
Dividing a fraction's numerator and denominator by a common factor reduces the fraction to lowest terms. 18/24 simplifies to 3/4 by dividing both numbers by their common factor of 6.
How do factors help simplify ratios?
The same common-factor division used for fractions also simplifies ratios. 18:24 simplifies to 3:4 by dividing both terms by their greatest common factor, 6.
How do factors connect to exponents?
Prime factorization often contains repeated prime factors, which are written more compactly using exponents. 2×2×2×3×3×5 becomes 2³×3²×5.
How do factors connect to square roots?
A perfect square has a factor pair where both factors are equal, and that repeated factor is the square root. Since 144's factor pair includes 12×12, √144 = 12.
Why do I only need to test factors up to the square root?
If both numbers in a factor pair were larger than the square root of n, their product would be larger than n itself, which is impossible. So at least one factor in every pair is at or below √n, and every factor beyond that point has already appeared as the partner of a smaller one.
How many factors does a number have?
Write the number's prime factorization, add 1 to each exponent, then multiply those results together. For 360 = 2³×3²×5, that is (3+1)(2+1)(1+1) = 4×3×2 = 24 factors.
How do you calculate the number of factors?
Use the divisor-count formula: if n = p1^a × p2^b × p3^c, the number of positive factors equals (a+1)(b+1)(c+1). This matches the length of the full factor list.
What divisibility rules help find factors?
Quick rules include: divisible by 2 if the last digit is even, by 3 if the digit sum is divisible by 3, by 5 if it ends in 0 or 5, and by 10 if it ends in 0, among others.
Is a factor the same as a scale factor?
No. An integer factor divides another integer evenly, while a geometric scale factor is a multiplier used to enlarge or reduce similar figures. They are related only by sharing the word "factor."
Can this calculator find scale factors?
No. This calculator finds integer factors, factor pairs, and prime factorizations. It does not compute geometric scale factors between similar figures.
Can this calculator factor polynomials?
No. This calculator factors integers, not algebraic expressions. Factoring a polynomial such as x²+5x+6 into (x+2)(x+3) requires algebraic methods, not integer division.
What is the difference between integer factoring and polynomial factoring?
Integer factoring breaks a whole number into integer divisors, such as 12 = 2×2×3. Polynomial factoring rewrites an algebraic expression as a product of simpler expressions, such as x²+5x+6 = (x+2)(x+3). They use related ideas but different methods.
Last reviewed: October 4, 2026