Factor Calculator
Find all factors of a positive integer, count them, sum them, and check if it is prime.
Result
What is a Factor Calculator?
A factor calculator finds every whole number that divides a given integer exactly — no remainder. Enter any positive integer up to 1 trillion and the tool lists its factors, counts them, sums them, and checks whether the number is prime — one whose only factors are 1 and itself.
Factoring comes up constantly: reducing fractions, finding greatest common factors and least common multiples, splitting quantities into equal groups, testing primality. By hand this gets tedious fast; the calculator applies the same square-root pairing method you would use on paper, instantly.
How the Factor Calculation Works
The method rests on one observation: factors come in pairs — if d divides n evenly, n ÷ d is also a factor — so you only need to test divisors up to √n.
d is a factor of n when n mod d = 0
For every d ≤ √n that divides n: both d and n ÷ d are factors
Count of Factors = total number of divisors found
Sum of Factors = 1 + d₁ + d₂ + ... + n
n is Prime when Count of Factors = 2 (only 1 and n itself)
- Number — the positive integer to factor, from 1 up to 1,000,000,000,000 (1e12).
- All Factors — every divisor, ascending, collected as pairs (d, n ÷ d) then sorted.
- Count of Factors — how many divisors exist; only perfect squares have an odd count (their square root pairs with itself).
- Sum of Factors — all divisors added together, including 1 and the number itself.
- Is Prime? — "Yes" only when the count is exactly 2.
Worked Example: Factoring 60
Take n = 60. The square root of 60 is about 7.746, so only the divisors from 1 to 7 need testing:
- 60 ÷ 1 = 60 exactly → factors 1 and 60
- 60 ÷ 2 = 30 exactly → factors 2 and 30
- 60 ÷ 3 = 20 exactly → factors 3 and 20
- 60 ÷ 4 = 15 exactly → factors 4 and 15
- 60 ÷ 5 = 12 exactly → factors 5 and 12
- 60 ÷ 6 = 10 exactly → factors 6 and 10
- 60 ÷ 7 = 8 remainder 4 → not a factor
Sorted, the full list is 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 — twelve factors. Adding them step by step: 1 + 2 = 3, + 3 = 6, + 4 = 10, + 5 = 15, + 6 = 21, + 10 = 31, + 12 = 43, + 15 = 58, + 20 = 78, + 30 = 108, + 60 = 168. Because the count is 12 rather than 2, 60 is not prime.
Cross-check with prime factorization: 60 = 2² × 3 × 5, and (2 + 1) × (1 + 1) × (1 + 1) = 12 — the same count.
For contrast, try 97: its square root is about 9.85, so only the primes 2, 3, 5, and 7 need testing, and none divides evenly (97 ÷ 2 = 48 remainder 1, 97 ÷ 3 = 32 remainder 1, 97 ÷ 5 = 19 remainder 2, 97 ÷ 7 = 13 remainder 6). The factors are just 1 and 97 — count 2, sum 98 — so 97 is prime.
Factor Counts and Sums for Common Numbers
Quick-reference results for frequently factored numbers:
| Number | Count of Factors | Sum of Factors | Prime? |
|---|---|---|---|
| 12 | 6 | 28 | No |
| 24 | 8 | 60 | No |
| 36 | 9 | 91 | No |
| 48 | 10 | 124 | No |
| 60 | 12 | 168 | No |
| 72 | 12 | 195 | No |
| 100 | 9 | 217 | No |
| 120 | 16 | 360 | No |
| 144 | 15 | 403 | No |
The perfect squares (36, 100, 144) have odd factor counts; every non-square has an even count — the pairing rule at work.
How to Use
- Type a positive integer from 1 to 1,000,000,000,000 into the Number field (whole numbers only).
- Click Calculate. Out-of-range or non-integer entries are flagged in red.
- Read the results: the complete factor list, the count, the sum, and the prime check.
- Click Reset to clear the field before trying another number.
Frequently Asked Questions
What is a factor?
A factor is a whole number that divides another number exactly, leaving no remainder. For example, 60 ÷ 4 = 15, so 4 is a factor of 60; but 60 ÷ 7 = 8 remainder 4, so 7 is not. Every positive integer has at least two factors: 1 and itself.
What is the largest number I can use?
This calculator supports positive integers up to 1 trillion (1,000,000,000,000, or 1e12). Even 1e12, which has 169 factors, resolves instantly.
How is the sum of factors calculated?
Every factor is added together, including 1 and the number itself. For 60: 1 + 2 + 3 + 4 + 5 + 6 + 10 + 12 + 15 + 20 + 30 + 60 = 168.
What is the difference between factors and prime factors?
Factors are all the divisors of a number; prime factors are only its prime building blocks. 60 has twelve factors, but its prime factorization is 2² × 3 × 5. Use a prime factorization calculator for that breakdown.
How do I find factors by hand?
Compute the square root of your number, then test each whole number up to it. Whenever one divides evenly, record both it and its pair (your number divided by that divisor). Testing beyond the square root only rediscovers pairs you already found.
Why does a perfect square have an odd number of factors?
Factors normally come in pairs: with 36, you get (1, 36), (2, 18), (3, 12), and (4, 9). The square root pairs with itself — 6 × 6 = 36 — and counts only once, giving 36 a total of 9 factors.
Is this calculator free?
Yes — completely free, and it runs entirely in your browser; nothing you enter is uploaded.