How to use the factor calculator
- Enter a positive integer in the Number field of the factor calculator.
- The tool instantly lists every factor, groups them into factor pairs, and shows the divisor count and sum.
- Use the result to simplify fractions, find common denominators or check if a number is prime (exactly two factors).
How the factor calculator works
The factor calculator finds all divisors by testing integers from 1 up to the square root of the input. For each divisor d that divides n evenly, both d and n/d are factors. This reduces the work from O(n) to O(√n). For example, with 60 the loop runs to 7 (since √60 ≈ 7.75). Divisors found: 1, 2, 3, 4, 5, 6. Their partners are 60, 30, 20, 15, 12, 10. Sorted, the full list is 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
Reference table of common values
Quick lookup for numbers often used in fraction work and scheduling.
| Number | Factors | Count | Sum |
|---|---|---|---|
| 1 | 1 | 1 | 1 |
| 2 | 1, 2 | 2 | 3 |
| 4 | 1, 2, 4 | 3 | 7 |
| 6 | 1, 2, 3, 6 | 4 | 12 |
| 12 | 1, 2, 3, 4, 6, 12 | 6 | 28 |
| 24 | 1, 2, 3, 4, 6, 8, 12, 24 | 8 | 60 |
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 | 9 | 91 |
| 48 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 | 10 | 124 |
| 60 | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 | 12 | 168 |
| 100 | 1, 2, 4, 5, 10, 20, 25, 50, 100 | 9 | 217 |
Practical uses for factor lists
Simplifying fractions: divide numerator and denominator by their greatest common factor. Finding common denominators: the least common multiple of two denominators uses their prime factorisations. Scheduling repeating events: a task every 6 days and another every 8 days align every 24 days (LCM of 6 and 8). Cryptography: RSA key generation relies on the difficulty of factoring large semiprimes. Game design: factor pairs determine rectangular grid layouts for a given tile count.
Common mistakes and edge cases
Zero and negative integers have no standard factorisation in elementary number theory; the factor calculator rejects them. Non-integers are rejected because divisibility is defined only for integers. Very large numbers (above 2^53 - 1) lose precision in JavaScript, so the tool caps input at Number.MAX_SAFE_INTEGER. Perfect squares produce a middle pair where both factors are equal (for example 6 × 6 for 36); this pair is listed once. Prime numbers always return exactly two factors: 1 and the number itself.
From factors to prime factorisation and beyond
The factor list is the starting point for prime factorisation. Write each composite factor as a product of primes until only primes remain. For 60: 60 = 2 × 30 = 2 × 2 × 15 = 2² × 3 × 5. The number of divisors can be computed directly from the prime exponents: if n = p₁^a₁ p₂^a₂ … then the divisor count is (a₁+1)(a₂+1)…. For 60 = 2² × 3¹ × 5¹ the count is (2+1)(1+1)(1+1) = 12. The sum of divisors uses the formula ∏ (pᵢ^(aᵢ+1)-1)/(pᵢ-1). For 60: (2³-1)/(2-1) × (3²-1)/(3-1) × (5²-1)/(5-1) = 7 × 4 × 6 = 168. Our GCF calculator and LCM calculator build on these ideas for two or more numbers.
Worked examples
| Example | Input | Result |
|---|---|---|
| Highly composite 60 | Number: 60 | Factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60; Factor pairs: 1 × 60, 2 × 30, 3 × 20, 4 × 15, 5 × 12, 6 × 10; Number of divisors: 12; Sum of divisors: 168 |
| Prime 97 | Number: 97 | Factors: 1, 97; Factor pairs: 1 × 97; Number of divisors: 2; Sum of divisors: 98 |
| Perfect square 144 | Number: 144 | Factors: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144; Factor pairs: 1 × 144, 2 × 72, 3 × 48, 4 × 36, 6 × 24, 8 × 18, 9 × 16, 12 × 12; Number of divisors: 15; Sum of divisors: 403 |
| Large number 1000000 | Number: 1000000 | Factors: 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 125, 160, 200, 250, 320, 400, 500, 625, 800, 1,000, 1,250, 1,600, 2,000, 2,500, 3,125, 4,000, 5,000, 6,250, 8,000, 10,000, 12,500, 15,625, 20,000, 25,000, 31,250, 40,000, 50,000, 62,500, 100,000, 125,000, 200,000, 250,000, 500,000, 1,000,000; Factor pairs: 1 × 1000000, 2 × 500000, 4 × 250000, 5 × 200000, 8 × 125000, 10 × 100000, 16 × 62500, 20 × 50000, 25 × 40000, 32 × 31250, 40 × 25000, 50 × 20000, 64 × 15625, 80 × 12500, 100 × 10000, 125 × 8000, 160 × 6250, 200 × 5000, 250 × 4000, 320 × 3125, 400 × 2500, 500 × 2000, 625 × 1600, 800 × 1250, 1000 × 1000; Number of divisors: 49; Sum of divisors: 2,480,437 |
FAQ
What is a factor of a number?
A factor (or divisor) of an integer n is an integer d that divides n with no remainder, so n = d × k for some integer k.
How do factor pairs work?
Every factor d pairs with n/d. For 12 the pairs are 1×12, 2×6, 3×4. In a perfect square like 36 the middle pair is 6×6, listed once.
How can I tell if a number is prime from the factor list?
A prime has exactly two factors: 1 and itself. If the factor calculator shows more than two factors, the number is composite.
Why does the tool reject zero and negative numbers?
Standard divisor definitions apply to positive integers. Zero has infinitely many divisors, and negative numbers mirror the positive factors with a sign change.
What is the largest number I can enter?
The maximum safe integer in JavaScript is 9,007,199,254,740,991 (2^53 - 1). Larger values may lose precision and give incorrect factor lists.
How are the divisor count and sum calculated?
The factor calculator enumerates all factors, then counts them and adds them. For large numbers the prime-factorisation formulas are faster, but enumeration is exact and clear for the range this tool handles.
Can I use this to find the greatest common factor of two numbers?
This tool factors one number at a time. For the GCF of two or more numbers, use the GCF calculator which computes it directly.