Factors Calculator

What this calculator does

Find every whole number that divides evenly into your number. Each one is called a factor (or divisor), because it goes in with nothing left over. For example, the factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. Factors come in pairs that multiply back to the number, such as 4 × 15 and 6 × 10. That makes them handy for arranging things in rows, splitting amounts evenly and simplifying fractions.

Enter a whole number from 1 up to 10^15 (a 1 followed by 15 zeros). First the calculator breaks it into primes (its prime factorization). A prime is a number with no factors except 1 and itself. From those primes, it builds every factor. Then it shows the full list, the factor pairs, how many factors there are and their sum. It also gives the sum of the proper factors, meaning all the factors except the number itself. Finally, it tells you whether the number is prime, perfect, abundant or deficient.

Where it's used

Why this math matters outside the classroom:

• Arranging chairs. A teacher with 24 chairs can set them out in equal rows of 1, 2, 3, 4, 6, 8, 12 or 24, the factors of 24.
• Equal teams. A coach with 36 players can split them evenly into teams of 2, 3, 4, 6, 9, 12 or 18.
• Garden plots. A gardener planning a 48 square foot rectangular bed can choose whole-number sides like 6 by 8 or 4 by 12.
• Packing cookies. A bakery packing 60 cookies into equal boxes can use boxes of 5, 6, 10, 12 or 15 with none left over.
• Shift planning. A manager splitting 30 hours into equal shifts can use 5 shifts of 6 hours or 6 of 5 hours, both factor pairs of 30.

Grade level and standards: Grade 4; Common Core State Standards for Mathematics 4.OA.B.4. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

1. Type a whole number between 1 and 1,000,000,000,000,000.
2. Read the number of factors at the top. The full list and the factor pairs come next.
3. Check the table for the factor pairs. Open the working to see how the count and sum come from the primes.

What the inputs mean

Whole number

Positive whole numbers only. A negative number has the same factors, with both signs: the factors of −12 are ±1, ±2, ±3, ±4, ±6 and ±12. So enter its size without the minus sign. Zero isn't allowed, because every whole number divides into it.

Formula and method

n = p₁^e₁ × p₂^e₂ × … × pₖ^eₖ
number of factors d(n) = (e₁ + 1)(e₂ + 1)…(eₖ + 1)
sum of factors σ(n) = ∏ (pᵢ^(eᵢ + 1) − 1) / (pᵢ − 1)
factor pairs: d × (n ÷ d) for every factor d ≤ √n

Here n is your number. The p's are its different primes, and each e (the exponent) is how many times that prime appears. k is how many different primes there are. d(n) is the number of factors, σ(n) (the Greek letter sigma) is their sum, and ∏ means multiply the brackets together. In the last line, d stands for any one factor.

Every factor of n is made by picking a power of each prime, from 0 up to that prime's exponent. For 60 = 2² × 3 × 5, you can pick 2⁰, 2¹ or 2² (three choices). You can pick 3⁰ or 3¹ (two) and 5⁰ or 5¹ (two). So there are 3 × 2 × 2 = 12 factors.

The sum works the same way. Multiplying out (1 + 2 + 4)(1 + 3)(1 + 5) gives every factor exactly once. Each bracket is a run of powers of one prime, and it has a shortcut: (p^(e+1) − 1)/(p − 1). For 60, that's 7 × 4 × 6 = 168.

Factors pair up as d and n ÷ d. One number in each pair is at most √n (the square root of n). So listing the small factors is enough to find them all. A perfect square has one pair whose two numbers are equal. That's why perfect squares are the only numbers with an odd number of factors.

Worked example

Find the factors of 60.

1. Factor it into primes: 60 = 2² × 3 × 5.
2. Count the factors: (2 + 1)(1 + 1)(1 + 1) = 12.
3. Pair them up: 1 × 60, 2 × 30, 3 × 20, 4 × 15, 5 × 12, 6 × 10.
4. Add them: σ(60) = (2³ − 1)/(2 − 1) × (3² − 1)/(3 − 1) × (5² − 1)/(5 − 1) = 7 × 4 × 6 = 168.
5. Leave out 60 itself: 168 − 60 = 108.

Result: 60 has 12 factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. They add up to 168. Its proper factors add up to 108, which is more than 60, so 60 is an abundant number.

Understanding the result

A number with exactly two factors (1 and itself) is prime. Any larger count means it's composite, meaning it can be split into smaller factors. The number 1 is the exception. It has a single factor, so it's neither prime nor composite.

Comparing the sum of the proper factors with the number sorts it into one of three groups. It's perfect when they're equal (6, 28, 496, 8,128). It's abundant when the sum is larger (12, 18, 20, 24). It's deficient when the sum is smaller (every prime, and every power of a prime).

The factor pairs are all the rectangles with whole-number sides that have that area. They're also the ways to split a group into equal rows.

Assumptions and limitations

• The input is a whole number from 1 to 10^15. That's small enough to factor exactly by trial division (testing one possible divisor after another).
• Lists with more than 300 factors are shortened in the middle. The pair table shows at most 150 rows. The count and sum always cover every factor.
• Only positive factors are listed.

Frequently asked questions

How this calculator is checked

Method: Divisors generated from the prime factorization; divisor-count and divisor-sum (σ) multiplicative formulas. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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