A factor of a whole number divides it with no remainder. Listing factors is a building block for simplifying fractions, finding common denominators, arranging objects into equal rows and checking whether a number is prime. This calculator lists every factor of a number up to 18 digits, groups them into factor pairs, highlights the prime factors and classifies the number by the sum of its factors.
How to use the factors calculator
- Enter a whole number. Commas are fine; a negative number is factored by its absolute value.
- The tape shows how many factors the number has, whether it is prime or composite, its prime factorization and the sums of its factors.
- Every factor appears as a chip, with prime factors highlighted, and the table lists each factor pair.
How to find factors by hand
The trick is that factors come in pairs, one small and one large, meeting at the square root. You only need to test numbers up to √n:
Worked example: factors of 120
Square root: √120 ≈ 10.95, so test 1 through 10.
Divides evenly: 1, 2, 3, 4, 5, 6, 8, 10 (7 and 9 leave remainders)
Pairs: 1 × 120, 2 × 60, 3 × 40, 4 × 30, 5 × 24, 6 × 20, 8 × 15, 10 × 12
All 16 factors: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
Reading the pairs down the left column and back up the right column puts the factors in order automatically.
Counting factors without listing them
If you know the prime factorization, you can count factors instantly: add 1 to each exponent and multiply. 120 = 2³ × 3 × 5, so it has (3+1)(1+1)(1+1) = 16 factors. This is how the calculator handles very large numbers — it factors into primes first with the same method as the prime factorization calculator, then builds every combination.
Divisibility shortcuts
| Divisible by | Test | Example |
|---|---|---|
| 2 | Last digit is even | 3,586 |
| 3 | Digit sum is a multiple of 3 | 471 → 4+7+1 = 12 |
| 4 | Last two digits form a multiple of 4 | 1,316 |
| 5 | Last digit is 0 or 5 | 2,745 |
| 6 | Divisible by both 2 and 3 | 522 |
| 9 | Digit sum is a multiple of 9 | 7,281 → 18 |
| 10 | Last digit is 0 | 6,930 |
Prime, composite, perfect, abundant, deficient
- A prime has exactly two factors, 1 and itself. A composite number has more than two. The number 1 has only one factor and is neither.
- Comparing a number with the sum of its proper factors (all factors except itself) sorts it into three families. Perfect numbers match exactly — 6, 28, 496 and 8,128 are the first four. Abundant numbers have proper factors that add up to more than the number, like 12 (1+2+3+4+6 = 16). Deficient numbers have proper factors that add up to less, like 10 (1+2+5 = 8) and every prime.
Where factors are useful
- Arrays and arrangements: 120 chairs can be set out as 8 rows of 15, 10 rows of 12, and so on — one arrangement for each factor pair.
- Fractions: knowing the factors of a numerator and denominator lets you reduce a fraction to lowest terms. The GCF calculator and common factors calculator compare factor lists for several numbers.
- Scheduling: a 120-minute block divides evenly into sessions of 15, 20, 24, 30, 40 or 60 minutes.
To check primality alone, use the prime number checker. To go in the other direction and list numbers that a value divides into, try the multiples calculator.
Frequently asked questions
How do I find all the factors of a number?
Test each whole number from 1 up to the square root of the number. Whenever one divides evenly, both it and the quotient are factors. For 120, the square root is about 10.95, and the divisors 1, 2, 3, 4, 5, 6, 8 and 10 give the pairs 1×120, 2×60, 3×40, 4×30, 5×24, 6×20, 8×15 and 10×12 — 16 factors in all.
What are factor pairs?
Two numbers that multiply to give the original number. The factor pairs of 28 are 1 × 28, 2 × 14 and 4 × 7.
Why do perfect squares have an odd number of factors?
Factors come in pairs, except when a number is paired with itself. In 36 = 6 × 6, the 6 counts only once, so 36 has 9 factors instead of an even number.
What is the difference between a factor and a multiple?
A factor divides into a number evenly; a multiple is the result of multiplying the number by a whole number. 4 is a factor of 12, and 12 is a multiple of 4.
What is a perfect number?
A number equal to the sum of its proper factors (all factors except itself). 28 is perfect because 1 + 2 + 4 + 7 + 14 = 28. Numbers whose proper factors add up to more are abundant (like 12), and those that add up to less are deficient (like 10).