Find every factor, all factor pairs and the prime factorization of a number — plus divisor sums and number properties.
The factors of a number are the whole numbers that divide it exactly, with no remainder. To find them, test each number from 1 up to the square root: whenever one divides evenly, both it and the quotient are factors. For example, 360 ÷ 12 = 30, so 12 and 30 are both factors. That pairing trick is why 360 has 24 factors but only needs 18 checks.
Enter a positive whole number from 1 to 1,000,000,000,000. Decimals and text are not accepted.
A factor (also called a divisor) of a whole number is any whole number that divides it exactly, leaving no remainder. The factors of 12 are 1, 2, 3, 4, 6 and 12, because each of those divides 12 evenly. 5 is not a factor, since 12 ÷ 5 leaves a remainder of 2.
Every whole number greater than 1 has at least two factors: 1 and itself. Numbers with exactly those two are prime; numbers with more are composite. The number 1 is a special case — it has just one factor, so it is neither prime nor composite.
What the letters mean. Here n is the number being factored, and a and b are a factor pair — two whole numbers that multiply together to make n. If a divides n, then b = n ÷ a is automatically a whole number, which is why factors always arrive two at a time.
Why the square root works. In every pair a × b = n, one member must be at or below √n and the other at or above it — because if both were larger than √n their product would exceed n. So checking up to √n finds every pair exactly once. The only time a pair repeats itself is when a = b, which happens for perfect squares such as 36 = 6 × 6.
Type a positive whole number and press Find Factors or the Enter key. You will get the full factor list, every factor pair, the prime factorization, and a table of properties including the divisor count, divisor sum and number classification. Tick Also show negative factors if you need them, tap an example button to try a preset number, or use Share link to copy a web address that reopens the page with your number already loaded.
The quickest reliable method is to work in pairs, up to the square root. Most people try to list factors in one long line and lose track; pairing keeps the job short and self-checking, because you stop the moment the two columns meet.
There are three reliable methods, and they complement each other.
Work upwards from 1, testing whether each number divides evenly. Divisibility rules make this fast — you can often tell at a glance. Stop once you reach the square root.
Write the pairs in two columns as you find them. For 36: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6. When the two columns meet, you have them all. This is the neatest way to be sure nothing is missing.
Split the number into any two factors, then keep splitting until only primes remain. For 360: 360 = 36 × 10 → (6 × 6) × (2 × 5) → 2³ × 3² × 5. The primes you end up with are the same no matter how you split — that is the Fundamental Theorem of Arithmetic.
2 if it ends in 0, 2, 4, 6 or 83 if its digits add to a multiple of 34 if the last two digits form a multiple of 45 if it ends in 0 or 56 if it passes both the 2 and 3 tests8 if the last three digits form a multiple of 89 if its digits add to a multiple of 910 if it ends in 011 if the alternating digit sum is 0 or a multiple of 11
A factor pair is two numbers that multiply to give the original. They are the most practical form of factors, because they answer real questions: how many equal rows, what rectangle dimensions, how to split a group evenly.
| Number | Factor pairs | How many factors |
|---|---|---|
| 12 | 1×12, 2×6, 3×4 | 6 |
| 16 | 1×16, 2×8, 4×4 | 5 |
| 24 | 1×24, 2×12, 3×8, 4×6 | 8 |
| 36 | 1×36, 2×18, 3×12, 4×9, 6×6 | 9 |
| 48 | 1×48, 2×24, 3×16, 4×12, 6×8 | 10 |
| 100 | 1×100, 2×50, 4×25, 5×20, 10×10 | 9 |
Notice that 16, 36 and 100 have an odd number of factors. That only happens for perfect squares, because one pair (4×4, 6×6, 10×10) is the same number twice and so counts once.
The numbers people look up most often, with their factors, prime factorization and factor count. Tap any number into the calculator above to see its full breakdown.
| Number | Factors | Prime factors | How many |
|---|---|---|---|
| 12 | 1, 2, 3, 4, 6, 12 | 2² × 3 | 6 |
| 24 | 1, 2, 3, 4, 6, 8, 12, 24 | 2³ × 3 | 8 |
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 | 2² × 3² | 9 |
| 48 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 | 2⁴ × 3 | 10 |
| 60 | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 | 2² × 3 × 5 | 12 |
| 72 | 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 | 2³ × 3² | 12 |
| 90 | 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90 | 2 × 3² × 5 | 12 |
| 100 | 1, 2, 4, 5, 10, 20, 25, 50, 100 | 2² × 5² | 9 |
| 120 | 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 | 2³ × 3 × 5 | 16 |
| 144 | 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144 | 2⁴ × 3² | 15 |
| 360 | 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360 | 2³ × 3² × 5 | 24 |
| 1000 | 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000 | 2³ × 5³ | 16 |
Notice that 144 has an odd factor count — it is a perfect square (12 × 12). Highly composite numbers like 360 and 120 have unusually many factors for their size, which is exactly why 360 was chosen for degrees in a circle.
| Feature | Factors | Multiples |
|---|---|---|
| Definition | Numbers that divide into n | Numbers made by multiplying n |
| Example for 12 | 1, 2, 3, 4, 6, 12 | 12, 24, 36, 48, … |
| How many | A limited set | Infinitely many |
| Size | Never larger than n | Never smaller than n |
| Used for | Simplifying fractions, GCF | Common denominators, LCM |
| Feature | All factors | Prime factors |
|---|---|---|
| Example for 12 | 1, 2, 3, 4, 6, 12 | 2 and 3 (as 2² × 3) |
| Includes 1? | Yes | No — 1 is not prime |
| Includes the number? | Yes | Only if the number is prime |
| Unique? | Yes, one fixed list | Yes — the Fundamental Theorem of Arithmetic |
Add up a number's proper factors (all factors except the number itself) and compare the total with the number:
The calculator reports which category your number falls into automatically.
Education. Factors underpin simplifying fractions, finding the GCF and LCM, and factorising algebraic expressions.
Arranging and packing. Factor pairs give every possible rectangular arrangement — seating plans, tiles per row, items per box, or team sizes that divide a group evenly.
Construction and design. Choosing a tile or panel size that divides a wall length exactly avoids cutting, which is a factor problem in disguise.
Scheduling. Working out when repeating events line up uses factors and multiples together.
Computing and cryptography. Prime factorization is easy to check but hard to reverse for very large numbers, and that asymmetry is the basis of widely used encryption such as RSA.
1) 1, 2, 3, 6, 9, 18 2) 1, 2, 3, 5, 6, 10, 15, 30 3) Composite (51 = 3 × 17) 4) 1×20, 2×10, 4×5 5) 3² × 5
1) (4+1)(2+1) = 15 2) Perfect (1+2+4+8+16+31+62+124+248 = 496) 3) It is a perfect square, so 7×7 counts once 4) 50 5) 2¹⁰
What is a factor?
A factor of a whole number is any whole number that divides it exactly, with no remainder. For example, 1, 2, 3, 4, 6 and 12 are all factors of 12.
How do you find all the factors of a number?
Test each whole number from 1 up to the square root. Whenever one divides evenly, both it and the quotient are factors. For 36 you test 1 to 6, giving the pairs 1x36, 2x18, 3x12, 4x9 and 6x6.
What is a factor pair?
A factor pair is two whole numbers that multiply together to make the original number. The factor pairs of 24 are 1x24, 2x12, 3x8 and 4x6.
What is prime factorization?
Prime factorization writes a number as a product of prime numbers only. For example, 360 = 2 x 2 x 2 x 3 x 3 x 5, usually written as 2 to the power 3, times 3 squared, times 5.
What is the difference between factors and multiples?
Factors divide into a number and are never larger than it, while multiples are made by multiplying the number and are never smaller. 12 has factors 1, 2, 3, 4, 6 and 12, and multiples 12, 24, 36 and so on forever.
What are proper factors?
Proper factors are all the factors of a number except the number itself. The proper factors of 12 are 1, 2, 3, 4 and 6.
How many factors does a number have?
Take the prime factorization, add 1 to each exponent and multiply the results. Since 360 = 2 cubed x 3 squared x 5, it has (3+1)(2+1)(1+1) = 24 factors.
Is 1 a prime number?
No. A prime number has exactly two different factors, but 1 has only one. That is why 1 is neither prime nor composite.
What is a perfect number?
A perfect number equals the sum of its proper factors. 6 is perfect because 1 + 2 + 3 = 6, and 28 is perfect because 1 + 2 + 4 + 7 + 14 = 28.
What are abundant and deficient numbers?
A number is abundant when its proper factors add up to more than the number, and deficient when they add up to less. 12 is abundant and 8 is deficient. Every prime number is deficient.
Why do perfect squares have an odd number of factors?
Factors normally pair up, giving an even count. In a perfect square one pair is the square root multiplied by itself, such as 6 x 6 = 36, so that factor is counted only once and the total becomes odd.
Can factors be negative?
Yes. Every positive factor has a negative counterpart, because two negatives multiply to a positive. The negative factors of 12 are -1, -2, -3, -4, -6 and -12. Tick the option in the calculator to display them.
What are the factors of 0?
Every whole number divides 0 exactly, so 0 has infinitely many factors. Because that cannot be listed, this calculator accepts numbers from 1 upwards.
How are factors used in real life?
Factors are used to simplify fractions, arrange items into equal rows or groups, choose tile and panel sizes that fit exactly, plan schedules, and in cryptography, where the difficulty of factoring very large numbers helps keep data secure.
Can factors repeat?
In a list of factors each number appears only once, so the factors of 12 are 1, 2, 3, 4, 6 and 12. Prime factors can repeat, however: 12 = 2 x 2 x 3, which is written as 2 squared times 3.
Can zero be a factor?
No. Dividing by zero is undefined, so zero is never a factor of any number. Zero itself is divisible by every whole number, which means it has infinitely many factors.
Can fractions be factors?
Not in ordinary factoring. Factors are whole numbers, so 12 has the factors 1, 2, 3, 4, 6 and 12. Fractions appear in algebra when factoring expressions, but not when listing the factors of a whole number.
Can decimals be factors?
No. A factor must divide a number exactly and leave a whole number, so decimals are not factors. If you enter a decimal, this calculator asks you for a whole number instead of silently rounding it.
What are common factors?
Common factors are the factors shared by two or more numbers. The factors of 12 and 18 share 1, 2, 3 and 6, so their greatest common factor is 6. Our GCF Calculator works this out for any pair of numbers.
Is this Factor Calculator free?
Yes. It is free with no sign-up, works on any device, and runs in your browser so your entries stay private.