About the Factor Calculator
This factor calculator lists every whole number that divides your number exactly, along with the factor pairs that multiply to give it. Type any whole number up to one billion and you instantly see the complete list of factors, how many there are, their sum, the prime factorization, and whether the number is prime.
It is handy for students simplifying fractions, finding common factors or checking multiplication facts, teachers preparing worksheets, and anyone arranging items into equal rows and columns — the factor pairs of 24 are exactly the rectangles you can make from 24 tiles. It also classifies the number as perfect, abundant or deficient by comparing it with the sum of its proper factors.
Factors here are positive divisors. Every negative number has the same factors with a minus sign added to one number of each pair, so for a negative input the calculator factors its absolute value and says so.
How to use the factor calculator
- 1Type a whole number between 1 and 1,000,000,000.
- 2Read the full list of factors in increasing order.
- 3Use the factor pairs to see which two numbers multiply to give it.
- 4Check the prime factorization and whether it is prime, perfect or abundant.
Formula and method
The calculator tests every whole number a from 1 up to √n; whenever a divides n exactly, both a and n ÷ a are factors and form a pair. Stopping at the square root is enough because the larger factor of every pair is above it, so even a number near one billion needs only about 31,600 checks.
The prime factorization gives a quick way to count factors: add one to each exponent and multiply. For 60 = 2² × 3 × 5 that is 3 × 2 × 2 = 12 factors. A number is perfect when its proper factors add up to the number itself (6, 28, 496), abundant when they add up to more, and deficient when less.
- n
- The number being factored
- pᵢ, eᵢ
- Prime factors of n and their exponents
- σ(n)
- Sum of all positive factors of n
Worked examples
Factors of 60
60 = 2² × 3 × 5, so it has (2+1)(1+1)(1+1) = 12 factors. They sum to 168, and the proper factors sum to 108, more than 60, so 60 is abundant.
Perfect number 28
The proper factors of 28 are 1, 2, 4, 7 and 14, which add up to exactly 28, making it a perfect number.
A prime number: 97
No whole number from 2 to √97 ≈ 9.8 divides 97, so its only factors are 1 and 97 and it is prime.
Perfect square 144
144 = 2⁴ × 3², giving (4+1)(2+1) = 15 factors. The count is odd because 12 × 12 pairs with itself.
Frequently asked questions
How do you find all the factors of a number?+
Divide the number by 1, 2, 3 and so on up to its square root. Each time the division is exact, write down both the divisor and the quotient. Together they make the complete list of factors.
What is the difference between factors and prime factors?+
Factors are all the whole numbers that divide a number exactly (60 has 12). Prime factors are only the prime numbers that multiply to make it (60 = 2 × 2 × 3 × 5). Every factor is a product of some of the prime factors.
What are factor pairs?+
Factor pairs are two numbers that multiply to give the original number. The factor pairs of 24 are 1 × 24, 2 × 12, 3 × 8 and 4 × 6, which also describe every rectangle you can make from 24 squares.
Why do perfect squares have an odd number of factors?+
Factors come in pairs a × b, except when a = b. A perfect square such as 36 has one pair where both factors are the same (6 × 6), so that factor is only counted once, giving an odd total.
What is a perfect number?+
A perfect number equals the sum of its proper factors. The first four are 6, 28, 496 and 8,128. Numbers whose proper factors add up to more are abundant (like 12), and less are deficient (like 8).