Factor Calculator
Find all factors, prime factorization, and factor pairs of any positive integer.
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List every factor of any positive integer, its prime factorization, and all factor pairs — plus whether the number is prime or perfect.
How Factor Calculator Works
The calculator finds every factor of your number by testing divisors from 1 up to its square root: whenever a candidate divides evenly, both that divisor and its matching pair (the number divided by it) are added to the list. Checking only up to the square root is enough because factors always come in pairs that multiply back to the original number.
Alongside the full factor list, it computes the prime factorization — breaking the number down into the unique combination of prime numbers that multiply together to produce it — by dividing out the smallest possible prime repeatedly until only primes remain.
From the factor list, the calculator also derives a few extra properties: whether the number is prime (exactly two factors, 1 and itself), whether it is a "perfect number" (its proper divisors add up to the number itself, like 6 = 1+2+3), and the sum of all its factors.
See It In Action
Who Uses Factor Calculator and Why
- Listing factor pairs to help factor a quadratic expression by hand.
- Checking whether a specific number is prime by seeing whether it has more than two factors.
- Splitting a total quantity — seats, boxes, students — into every possible arrangement of equal rows and columns.
- Checking whether a number is a rare "perfect number" for a math enrichment exercise.
Mistakes to Avoid
- Treating the full factor list and the prime factorization as the same thing — factors include every divisor (like 1, 4, 6, 9, and 360 itself for 360), while prime factorization is only the specific prime building blocks (2³ × 3² × 5).
- Assuming a large number automatically has few factors — a highly composite number like 360 has 24 factors despite not being especially large.
- Overlooking that the calculator's square-root shortcut only needs to test divisors up to √n, so a slow result for a very large number reflects genuine computation, not a bug.
Tips for Best Results
- Use the factor-pair list, not just the flat factor list, when you specifically need ways to write the number as a product of two whole numbers.
- Cross-check primality quickly: if the factor list contains anything besides 1 and the number itself, it's composite, not prime.
Fixing Common Problems
I'm not sure if my number is prime from the results. — Count the factors listed — exactly two (1 and itself) means prime; more than two means composite.
The factor sum looks too large to be a perfect number. — Perfect numbers require the sum of proper divisors only (excluding the number itself) to equal the number exactly — check whether the calculator's total includes the number itself before comparing.
Terms Explained
Factor pair: Two factors that multiply together to produce the original number.
Perfect number: A number whose proper divisors (all factors except itself) sum exactly to the number, like 6 (1+2+3) or 28.