Factor Calculator

The Factor Calculator lists all positive integer divisors of a number — the complete factor list — plus the prime factorization in compact exponent form. Used in number-theory homework, simplification of fractions, finding the greatest common factor of related numbers, and curiosity ("what are the factors of 360?"). Trial-divides up to √N (the only divisors past that are pairs with the ones below); fast enough for inputs up to about 10^12.

Built by Bob Article by Lace QA by Ben Shipped May 12, 2026

How to use

1
Enter a positive integer. The calculator accepts up to 10^12 (one trillion) — beyond that, trial division becomes slow.
2
Read the complete factor list — everything that divides evenly into N, sorted smallest to largest. A prime number has exactly 2 factors (1 and itself); a perfect square has an odd number of factors.
3
Read the prime factorization below — the unique decomposition into prime powers. Every positive integer has exactly one prime factorization (the Fundamental Theorem of Arithmetic).
4
The properties line shows whether N is prime, a perfect square, even, or odd.

÷ Greatest Common Factor Calculator × Least Common Multiple Calculator √ Root Calculator x² Exponent Calculator

Frequently asked questions

What's a "factor"?

A factor (or divisor) of N is any positive integer that divides N evenly — leaves no remainder. 1 and N itself are always factors. Other factors come in pairs that multiply to N: if 12 = 3 × 4, then 3 and 4 are both factors of 12. 12's complete factor list is 1, 2, 3, 4, 6, 12 — six factors total.

What's prime factorization?

Decomposing N into the product of prime numbers raised to powers. Every positive integer >1 has a unique prime factorization (this is the Fundamental Theorem of Arithmetic — Euclid proved it ~300 BC). Example: 360 = 2³ × 3² × 5 (because 360 = 8 × 9 × 5 = 2·2·2 · 3·3 · 5). The exponents and primes are unique up to order. Useful for: finding GCD/LCM, simplifying fractions, cryptography (RSA security relies on the difficulty of factoring large numbers).

How do I find factors by hand?

Trial division. Start at 1 and divide N by each integer up to √N. If the division is exact (no remainder), you've found a factor pair: i and N/i. Stop at √N because any factor pair has one element below √N and one above. Example for N=36: try 1 (✓, pair 36), 2 (✓, pair 18), 3 (✓, pair 12), 4 (✓, pair 9), 5 (✗), 6 (✓, pair 6 — same as itself, perfect square). Full list: 1, 2, 3, 4, 6, 9, 12, 18, 36.

How is prime factorization computed?

Trial division by primes only. Start with the smallest prime (2): divide N by 2 as many times as possible, count the divisions = exponent of 2. Move to next prime (3), repeat. Continue through 5, 7, 11, etc. up to √(remaining N). When you can't divide further, whatever's left (if > 1) is itself a prime factor. Example for 360: 360/2=180/2=90/2=45 → 2³. 45/3=15/3=5 → 3². 5 is prime → 5¹. Result: 360 = 2³ × 3² × 5.

What's a prime number?

A positive integer >1 whose only factors are 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29... There are infinitely many (Euclid proved this too). 1 is not prime by modern convention — it has only one divisor, not two. 2 is the only even prime; every other prime is odd. The calculator flags N as prime if its factor list has exactly 2 entries (1 and itself).

Why are big numbers hard to factor?

Trial division runs in O(√N) time. For N = 10^9, √N = ~31,623 — fast. For N = 10^12, √N = ~10^6 — still fast (under 1 second). For N = 10^20 (the size of typical RSA modulus components), √N = 10^10 — would take hours. For real RSA-2048 (N ~ 10^617), trial division would take longer than the universe has existed. This calculator caps at 10^12 to keep response time under a second; specialized algorithms (Pollard rho, quadratic sieve, GNFS) scale better but are out of scope for a browser tool.

What's the relationship between factors of N and prime factorization?

Every factor of N is a product of some subset of the primes in N's prime factorization. Example: 360 = 2³ × 3² × 5 has (3+1)(2+1)(1+1) = 24 factors total — multiply (each prime's exponent + 1) across the factorization. The 24 factors of 360 are every combination of 2^a × 3^b × 5^c where 0 ≤ a ≤ 3, 0 ≤ b ≤ 2, 0 ≤ c ≤ 1. The number-of-divisors formula τ(N) = ∏(eᵢ + 1).

Why do perfect squares have an odd number of factors?

Because factors come in pairs that multiply to N — except when both elements of the pair are the same (the square root). For N = 36, the pair (6, 6) counts once instead of twice, so the total is odd: 1, 2, 3, 4, 6, 9, 12, 18, 36 → 9 factors. For any non-square N, every factor below √N has a distinct partner above √N, so the count is even. The calculator detects perfect squares and tells you so when it applies.

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What are factors?

A factor of a number N is any positive integer that divides N with no remainder. 12 has six factors: 1, 2, 3, 4, 6, and 12. Each one divides 12 cleanly — 12 ÷ 3 = 4 with nothing left over. Factors are sometimes called divisors; the words mean the same thing.

Every positive integer has at least two factors — 1 and itself. Prime numbers have exactly those two, no others. Composite numbers have more. The number of factors a number has tells you something about its structure: prime numbers have 2, the squares of primes have 3 (1, p, p²), and highly composite numbers like 360 or 5040 can have dozens.

Factors come in pairs that multiply to N. For 36: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6). Notice the last pair is the same number twice — that happens when N is a perfect square. The pairing is why you only have to test divisors up to √N when finding factors by hand. Anything past the square root is just the larger half of a pair you've already discovered.

How to use the Factor Calculator

The Factor Calculator takes a positive integer and returns three pieces of information: every factor in order, the prime factorization in compact exponent form, and a properties line (prime, perfect square, even, odd). The result updates live as you type — no button to press, nothing sent to a server.

Enter a positive integer. The tool accepts inputs up to about 10¹² (one trillion).
Read the complete factor list, smallest to largest.
Read the prime factorization beneath it (for example, 360 = 2³ × 3² × 5).
Check the properties line — flags for prime, perfect square, even, and odd.

If you're doing fraction arithmetic, the prime factorization is the quickest path to a common denominator. If you're working through number-theory exercises, the factor count itself is often the answer. If you're just curious — "how many ways can I arrange 60 chairs in equal rows?" — the factor list is the answer to that, too.

Trial division — how the calculator finds factors

The Factor Calculator uses trial division, the same method you'd use by hand but faster. Walk through every integer from 1 to √N. If it divides N exactly, you've found a factor — and its pair (N divided by it) is also a factor. Collect both, move on.

You only need to check divisors up to √N. Every factor above √N is automatically paired with one below.

For N = 100, √N = 10. Test 1 through 10. The divisors found are 1, 2, 4, 5, 10. Their pairs (100, 50, 25, 20, 10) fill in the rest. Sort and deduplicate: 1, 2, 4, 5, 10, 20, 25, 50, 100. Nine factors total — 100 is a perfect square, so the count is odd.

For a 12-digit input, √N is around a million. Testing a million candidates with a tight loop takes well under a second in JavaScript. Beyond 10¹², trial division starts to feel slow in the browser, which is why that's the cap.

Worked example: factors of 36

36 is the kind of number that shows up in homework constantly because it has nine factors — a generous variety without being unwieldy. Walk through it:

√36 = 6. Test divisors 1 through 6.
1 divides 36 → factor pair (1, 36).
2 divides 36 (36 ÷ 2 = 18) → factor pair (2, 18).
3 divides 36 (36 ÷ 3 = 12) → factor pair (3, 12).
4 divides 36 (36 ÷ 4 = 9) → factor pair (4, 9).
5 does not divide 36 (36 ÷ 5 = 7.2) → skip.
6 divides 36 (36 ÷ 6 = 6) → factor pair (6, 6). Same number both sides — 36 is a perfect square.

Collected and sorted: 1, 2, 3, 4, 6, 9, 12, 18, 36. Nine factors. The factor count is odd because of the (6, 6) self-pair — perfect squares always have an odd factor count for the same reason.

The prime factorization of 36 is 2² × 3². Two 2s and two 3s, no other primes. Notice the factor count (9) equals (2 + 1) × (2 + 1) — that's the number-of-divisors formula τ(N) = ∏(eᵢ + 1), where the eᵢ are the exponents in the prime factorization. It works for every positive integer. For 360 = 2³ × 3² × 5, the factor count is (3+1)(2+1)(1+1) = 24.

Factors of common numbers

Some numbers come up so often in fractions, geometry, and time arithmetic that knowing their factor lists by heart pays off. 60 shows up in time (60 seconds, 60 minutes) precisely because the Babylonians picked it for its factor richness — 12 factors total. 360 shows up in degrees for the same reason — 24 factors.

N	Factors	Count	Prime factorization
12	1, 2, 3, 4, 6, 12	6	2² × 3
24	1, 2, 3, 4, 6, 8, 12, 24	8	2³ × 3
36	1, 2, 3, 4, 6, 9, 12, 18, 36	9	2² × 3²
60	1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60	12	2² × 3 × 5
100	1, 2, 4, 5, 10, 20, 25, 50, 100	9	2² × 5²
144	1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144	15	2⁴ × 3²
360	1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360	24	2³ × 3² × 5

Notice that 36, 100, and 144 all have odd factor counts. They're the perfect squares in the table (6², 10², 12²). Every perfect square has this property, and only perfect squares do.

Prime factorization — the unique decomposition

Every positive integer above 1 can be written as a product of prime numbers raised to powers, and that decomposition is unique up to ordering. This is the Fundamental Theorem of Arithmetic, proved by Euclid around 300 BC. Saying it formally: for any N > 1, there are unique primes p₁ < p₂ < … < p_k and positive exponents e₁, e₂, … e_k such that N = p₁^e₁ × p₂^e₂ × … × p_k^e_k.

To find the prime factorization by hand, divide by the smallest prime that fits, count how many times it fits, move on to the next prime. For 360:

360 ÷ 2 = 180. 180 ÷ 2 = 90. 90 ÷ 2 = 45. Three divisions by 2. So 2³.
45 ÷ 3 = 15. 15 ÷ 3 = 5. Two divisions by 3. So 3².
5 is prime. Done. So 5¹.
360 = 2³ × 3² × 5

The Factor Calculator does this automatically and displays the result in standard exponent notation. The relationship between the prime factorization and the full factor list is direct: every factor of N is a product 2^a × 3^b × 5^c where 0 ≤ a ≤ 3, 0 ≤ b ≤ 2, 0 ≤ c ≤ 1 — every combination of the exponents, giving 4 × 3 × 2 = 24 factors total.

Why factorization matters

If you're stuck on a factorization problem, it's usually because the practical use feels abstract. Here's why it shows up:

Simplifying fractions. 48/36 simplifies to 4/3 by dividing both sides by their GCF (12). The GCF comes from comparing prime factorizations: 48 = 2⁴ × 3, 36 = 2² × 3², so GCF = 2² × 3 = 12. The Greatest Common Factor Calculator does this directly.
Finding a common denominator. Adding 1/12 + 1/18 needs LCM(12, 18) = 36. LCM comes from taking the maximum exponent of each prime — 12 = 2² × 3, 18 = 2 × 3², so LCM = 2² × 3² = 36.
Cryptography. RSA security relies on the fact that factoring large numbers is hard. For 200-digit inputs, even the fastest known algorithms take years on dedicated hardware.
Music and rhythm. The richness of 60 and 360 as factor sets is why time and angle measurement use them. Easy to divide into halves, thirds, quarters, fifths, sixths, tenths, twelfths.
Scheduling. If event A repeats every 12 days and event B every 18 days, the next time they coincide is in LCM(12, 18) = 36 days. Factor analysis answers this.

Related tools

The Factor Calculator is the workhorse for the rest of the number-theory family:

Prime Number Checker — quick yes/no on whether a single number is prime, without the full factor list.
Greatest Common Factor Calculator — finds the GCF of any list of numbers via the Euclidean algorithm.
Least Common Multiple Calculator — partners with GCF; LCM × GCF = a × b for any pair.
Fraction Simplifier — divides numerator and denominator by their GCF to get lowest terms.
Square Root Calculator — useful when you're trial-dividing by hand and need to know where to stop.

Frequently asked questions

What's the difference between a factor and a prime factor?

A factor is any positive integer that divides N. A prime factor is a factor that's also prime. 12 has factors 1, 2, 3, 4, 6, 12; its prime factors are 2 and 3. The prime factorization (12 = 2² × 3) lists the prime factors along with how many times each one divides N.

Why does the calculator say 1 is a factor of every number?

Because it is. 1 divides every positive integer with no remainder (N ÷ 1 = N). By the standard definition of factor, 1 is always included. So is N itself. The factor list is bracketed by these two — everything else is between them.

Is 1 a prime factor?

No. 1 is a factor of every number, but 1 is not prime. Prime factors must be prime numbers (2 or larger, with no divisors other than 1 and itself). The prime factorization of 12 is 2² × 3, not 1 × 2² × 3 — including 1 would let you write the same number infinitely many ways and break the uniqueness of the factorization.

Why do some numbers have so many more factors than others?

Because of how their prime factorizations look. A number with several small primes raised to moderate powers (like 360 = 2³ × 3² × 5) has many factors. A number that's a large prime (like 997) has only 2. The number-of-divisors formula τ(N) = ∏(eᵢ + 1) lets you compute the factor count from the prime factorization without listing the factors. Highly composite numbers (1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, …) are numbers with more factors than any smaller positive integer.

How does the calculator handle very large numbers?

It caps at about 10¹² (one trillion). Below that, trial division up to √N takes under a second in the browser. Above that, you'd notice the lag, and JavaScript's 53-bit integer precision starts to matter. For genuinely huge inputs (say, the 600-digit numbers used in RSA), no browser-side approach works — specialized algorithms (Pollard rho, quadratic sieve, GNFS) handle those, but they're heavy enough to require dedicated hardware.

What's a perfect square and why does the calculator flag it?

A perfect square is a number that equals some integer multiplied by itself — 1, 4, 9, 16, 25, 36, 49, 64, … Perfect squares always have an odd number of factors, because one factor pair is the square root with itself. The flag is useful in fraction simplification (perfect squares always reduce to integers under a square root) and number-theory exercises.

Can negative numbers have factors?

Yes, but conventionally factor lists are written using positive integers only. -12 has the same factor list as 12 (1, 2, 3, 4, 6, 12), with the understanding that each factor could be paired with either +N/factor or -N/factor. The Factor Calculator takes the absolute value of negative inputs to keep the output clean.

Why is prime factorization unique?

This is the Fundamental Theorem of Arithmetic, proved by Euclid around 300 BC. The short version: suppose a number N had two different prime factorizations. Cancel any primes they share. What's left would be two different products of distinct primes that equal each other — and you can prove via Euclid's Lemma (if a prime divides a product, it divides one of the factors) that this is impossible. The uniqueness is what makes factorization useful as a tool. If 12 could be written as 2² × 3 or as something else entirely, fraction arithmetic would be a mess.