Factor Calculator
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About Factors
Factoring is arithmetic's disassembly manual: homework (simplifying fractions, factoring quadratics), practical splitting (dividing 60 students into equal teams — the factor list IS the option list), and number theory's favorite pastime (primes, perfect numbers) all start from the same list this tool produces.
Enter any whole number up to 10 million for the complete factor list, count, prime factorization, and factor pairs. Two-factor results get the prime badge; the pairs view doubles as a 'ways to arrange in a rectangle' answer for everything from classroom seating to tile layouts.
Comparing factors ACROSS two numbers? That's the GCF Calculator
The Square-Root Method
Why nobody checks all the way up to n:
For i from 1 to √n: if n mod i = 0 → i and n÷i are both factors Factor count from primes: (e₁+1)(e₂+1)… for n = p₁^e₁ × p₂^e₂…
Worked example: 60 needs checks only to 7 (√60 ≈ 7.7), yielding pairs 1×60, 2×30, 3×20, 4×15, 5×12, 6×10 — twelve factors, matching the exponent formula on 2²×3×5: (2+1)(1+1)(1+1) = 12. ✓
Worked Examples
Numbers people actually look up — computed by this calculator:
| Number | Factors | Prime factorization | Count |
|---|---|---|---|
| 24 | 1, 2, 3, 4, 6, 8, 12, 24 | 2³ × 3 | 8 |
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 | 2² × 3² | 9 |
| 60 | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 | 2² × 3 × 5 | 12 |
| 97 | 1, 97 | 97 (prime) | 2 |
| 144 | 15 factors, 1 through 144 | 2⁴ × 3² | 15 |
Odd factor counts (36's nine, 144's fifteen) mark perfect squares — their square root pairs with itself, breaking the even-pairs pattern.
Divisibility Shortcuts
The classics that speed hand-factoring: 2 (even last digit), 3 (digit sum divisible by 3), 4 (last two digits divisible by 4), 5 (ends in 0/5), 6 (rules for 2 and 3 both), 9 (digit sum divisible by 9), 10 (ends in 0). For 11: alternately add and subtract digits — divisible if the result is. Combined, they crack most everyday numbers before a calculator wakes up.
Prime factorization's quiet superpowers: any factor of n is a product of its prime factors taken with equal-or-smaller exponents (which is how the count formula works), GCF takes the shared primes at minimum exponents while LCM takes all primes at maximum, and cryptography's foundation is precisely that factoring VERY large numbers stays hard — your number here surrenders in microseconds; a 600-digit one guards bank transfers.
Frequently Asked Questions
What are the factors of 60?
1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 — twelve of them, from the factorization 2² × 3 × 5. The pairs (2×30, 4×15, 6×10…) are the equal-group splits of 60.
What's the difference between factors and multiples?
Factors divide INTO the number (finite list: 12's are 1,2,3,4,6,12); multiples are the number times 1, 2, 3… (infinite: 12, 24, 36…). Factors ≤ the number, multiples ≥ it — the two directions of the same times table.
How do I find factors quickly by hand?
Test divisors up to the square root only, banking both halves of each hit: for 96, checking 1–9 finds everything (1×96, 2×48, 3×32, 4×24, 6×16, 8×12). The divisibility shortcuts above prune the tests further.
What is prime factorization used for?
It's the master key: reconstructing all factors, computing GCF/LCM (shared vs combined primes), simplifying radicals (√72 = √(2³×3²) = 6√2), and — scaled up brutally — the mathematical padlock behind modern encryption.
Is 1 a prime number?
No — primes need exactly two distinct factors, and 1 has only itself. The convention keeps prime factorization unique (otherwise 60 = 1×1×2²×3×5 = … forever). 2 is the smallest prime and the only even one.
How can I tell if a big number is prime?
Trial division to the square root settles anything this calculator accepts (enter it — two factors = prime). At cryptographic sizes, probabilistic tests (Miller–Rabin) answer in milliseconds what trial division couldn't finish before the sun burns out.
Methodology. This calculator uses standard, peer-reviewed mathematical formulas. It is reviewed and maintained by the Vast Calculators editorial team.
Last updated · July 11, 2026
Results are estimates for general use; verify critical figures independently.