Factor Calculator

🧮What are factors, really?

A factor of a number is a whole number that divides it with no remainder. The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72 — twelve in total, adding up to 195. (Mathematicians also say 72 is “divisible by” each of these, or that each is a “divisor” of 72 — the three words mean the same thing.)

Factors always arrive in factor pairs: if d divides n, then n ÷ d is the partner. For 72 the pairs are (1, 72), (2, 36), (3, 24), (4, 18), (6, 12) and (8, 9). Notice the pairs “meet in the middle” — and because of that, you only ever need to search up to √n to find them all, which is exactly what the calculator above does.

Two facts that never change: 1 is a factor of every whole number, and every whole number is a factor of itself. So any number n ≥ 1 has at least the factor pair (1, n).

Stated as a definition: what is a factor of a number? In factor math, a factor is a whole number that multiplies with another whole number to produce the number you started with. 4 is a factor of 20 because 4 × 5 = 20; 6 is not, because no whole number multiplied by 6 gives 20. That multiplication picture is the whole of factor in math — whenever a × b = n, both a and b are factors of n, and every factor of n arises that way.

The definition also settles the edge cases. 1 is a factor of every number (1 × n = n), and n is always a factor of itself, which is why every factor list on this page opens with 1 and closes with the number itself. A prime is the limiting case: 13 = 13 × 1 and no other multiplication, so its factor list is exactly two long.

🧩Factor pairs: how I find factors by hand

Whenever I explain how to find factors of a number, I start with pairs, not with the full list. A factor pair is two factors that multiply to give the number: find one and you have automatically found the other, because n ÷ d is always waiting on the other side of the multiplication. It is the single biggest shortcut in this whole topic, and it is the reason factoring by hand is much faster than it looks.

Watch it work on 48. I test small divisors in order and write down both partners: 1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8. Five and 7 do not divide 48, and by 6 × 8 the partners have nearly met — √48 ≈ 6.9, so once my trial divisor passes that point I only re-meet pairs I already own. Sorting the partners gives the full list: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 — ten factors.

This is exactly what the calculator above prints, and it is why I read the factor pairs first whenever I work a number by hand: the pairs show the structure, and the sorted list is just the pairs laid flat. One warning from experience — when the two partners are equal, as in 6 × 6 = 36, the pair counts once. That single fact explains every odd factor count you will ever meet, including 36’s nine factors.

🔍Prime factorization and the fundamental theorem

A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 …). Prime factorization breaks any whole number above 1 into a product of primes — its indivisible building blocks. For 360:

360 = 36 × 10 = (2² × 3²) × (2 × 5) = 2³ · 3² · 5. In other words, the prime factors of 360 are 2, 2, 2, 3, 3 and 5.

The exponents do more than look tidy — they encode the whole factor structure. The number of factors is (3+1)(2+1)(1+1) = 24, and the factor sum is ((2⁴−1) ÷ (2−1)) × ((3³−1) ÷ (3−1)) × ((5²−1) ÷ (5−1)) = 15 × 13 × 6 = 1,170 — matching the calculator exactly.

The fundamental theorem of arithmetic guarantees this breakdown is unique: every whole number above 1 has exactly one prime factorization (up to ordering). That uniqueness is why prime factorization is the standard fingerprint of a number — two numbers are equal if and only if their prime factorizations match.

Try the prime factorization calculator above with any number to see its unique breakdown instantly. Another example: 100 = 2² · 5², so it has (2+1)(2+1) = 9 factors (1, 2, 4, 5, 10, 20, 25, 50, 100) summing to 217. Note that 10 × 10 is a pair with itself — a square number always has an odd factor count.

Two methods, one answer. If you are learning how to do prime factorization by hand, the factor tree and the ladder are the two standard routes — here is how to find prime factorization both ways on the same number, 360.

The factor tree. Split 360 into any two factors and keep splitting the composite branches: 360 = 36 × 10; 36 = 6 × 6 = (2 × 3) × (2 × 3); 10 = 2 × 5. Every branch ends in a prime, and collecting the leaves gives the product of prime factors: 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ · 3² · 5.

The ladder. Divide by the smallest prime that fits, divide the answer, and keep going until the answer is 1:

The divisors used — 2, 2, 2, 3, 3, 5 — are the prime factorization, in the order they were used.

Both routes land on 2³ · 3² · 5, and they must: the fundamental theorem says the breakdown is unique. Which raises the definition underneath both methods — what are prime factors? They are the factors of a number that are themselves prime: the prime factors of 360 are 2, 3 and 5, each repeated as the exponents record. Every other factor of 360 — all 24 of them, from 1 to 360 — is built by multiplying these primes together in some combination. And that is how to find the prime factorization of a number too large for a tidy tree: ladder it, then read the divisors.

🔬Prime factors: reading the second answer the tool gives

I read the result card above as two answers, not one. The chips are the factor list; the headline is the prime factorization — in effect a prime factors calculator view of the same number, because the headline names the primes the number is built from. Take 84: its prime factors are 2, 2, 3 and 7, written compactly as 2² · 3 · 7. Said with repetition, those are the prime factors with multiplicity; said once — 2, 3, 7 — they are the distinct prime factors.

The distinction matters because the two lists answer different questions. The full factor list of 84 has twelve entries (1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84) and tells me every way 84 splits evenly. The prime factors tell me what 84 is made of — and I can rebuild every one of those twelve factors from them by switching each prime on or off up to its exponent: 2 can appear 0, 1 or 2 times, while 3 and 7 appear 0 or 1 times. That is exactly where the factor count (2+1)(1+1)(1+1) = 12 comes from.

My habit: when a problem asks what a number is made of — simplifying a fraction, finding a GCF, testing whether it is a perfect square — I read the prime factors first. When it asks how a number can be split, shared or arranged into equal groups, I read the factor list. Same number, two lenses; the tool shows both because real problems need both.

⭐Prime, composite, perfect — the special families

Factors sort every whole number into a family. A prime (like 97) has exactly two factors. A composite (like 60 or 72) has more than two. And 1 sits alone: with a single factor, it is neither prime nor composite.

Then there are the perfect numbers — numbers equal to the sum of their proper factors (all factors except the number itself):

When the proper-factor sum exceeds the number, it is called abundant (12 is the smallest); when it falls short, deficient — which includes every prime. Only four perfect numbers are below 10,000 (6, 28, 496 and 8,128), and finding them is still an active area of number theory.

✋Finding factors by hand

The calculator uses trial division, and you can do the same on paper: test each d from 1 up to √n, and keep the pairs. Divisibility rules let you skip most candidates:

Worked by hand: 144. It is even (2 × 72), digit sum 9 (3 × 48), ends in 44 (4 × 36) … continuing to √144 = 12 gives factors 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144 — 15 factors summing to 403, since 144 = 2⁴ · 3² gives (4+1)(2+1) = 15.

Two of these rules deserve a note. The 8 rule only looks at the last three digits because 1,000 is itself divisible by 8, so everything above the hundreds place can be ignored. And the 12 rule is a combination rule: since 3 and 4 share no factor, passing both tests separately guarantees divisibility by their product, 12. The calculator runs the full trial division for you, but with this chart I can usually name a number’s small factors before I have finished reading it.

Here is the sweep as a recipe — how to find the factors of a number, start to finish, on 60. Its square root is about 7.7, so only the divisors 1 to 7 need testing: 1 (pair 60), 2 (pair 30), 3 (pair 20), 4 (pair 15), 5 (pair 12), 6 (pair 10), and 7, which leaves a remainder and is skipped. Six pairs, twelve factors — 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 — and the moment the trial divisor passes the square root, every partner has already appeared, so the list is provably complete. The divisibility chart above is a shortcut for the same tests: it tells you which small divisors are worth dividing by at all.

🔗What factors are actually good for

Simplifying fractions. To reduce 36/48, find the greatest common factor (GCF) of 36 and 48. Prime factorizations: 36 = 2² · 3², 48 = 2⁴ · 3. Take the lowest power of each shared prime: 2² · 3 = 12. Divide top and bottom by 12: 36/48 = 3/4.

GCF and LCM of any pair. For 48 and 60: 48 = 2⁴ · 3, 60 = 2² · 3 · 5. The GCF takes lowest shared powers: 2² · 3 = 12. The least common multiple (LCM) takes highest powers: 2⁴ · 3 · 5 = 240. Check: 48 × 60 = 2,880 = 12 × 240 — always true.

Splitting things evenly. Can 96 items be packed into equal boxes of more than one? 96 = 2⁳ · 3 has 12 factors, so yes — boxes of 2, 3, 4, 6, 8, 12, 16, 24, 32 or 48 all work.

Cryptography. Multiplying two large primes is easy; recovering them from the product — factoring — is extraordinarily hard. That asymmetry secures RSA encryption. In 2009, researchers factored RSA-768, a 232-digit number, and it took roughly two years of computing time spread across hundreds of machines. Your browser factors a trillion in milliseconds; 232 digits is a different universe.

🧭Greatest common factor: reading it off two lists

When I need the greatest common factor of two numbers, I do not reach for a formula first — I put the two factor lists side by side. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. The entries appearing in both lists — the common factors — are 1, 2, 3, 4, 6 and 12, and the greatest of them, 12, is the GCF.

That comparison is the definition, and it agrees with the prime-factor shortcut in the uses section above (lowest shared powers: 2² · 3 = 12). It is also honest about scope: this page factors one number at a time, so for a GCF I run the tool twice and compare the lists. A dedicated two-number tool for this is on my list of pages to build; until then, the two lists plus thirty seconds of comparison do the job.

🔀Factors and multiples, side by side

I keep factors and multiples straight with one picture: factors live inside a number, multiples live outside it. Here is the comparison I wish someone had shown me first, using 12:

The confusion survives because the two ideas share one multiplication: 3 × 4 = 12 makes 3 and 4 factors of 12, and makes 12 a multiple of both. One equation, two directions — factors and multiples are the same fact read from opposite ends.

🧰Factors, GCF and LCM — which tool when

Three tools on this site divide the work, and picking the right one is mostly about how many numbers you start with. One number, and you want everything it is made of — every factor, the pairs, the prime factorization: that is this page. Two numbers, and you want the largest factor they share: that is the GCF, and it has its own tool, the GCF Calculator. Two numbers, and you want the smallest multiple they share — the first number both divide into: that is the job of the LCM Calculator.

The three answers are built from the same raw material. The GCF of 48 and 60 is 12 because 12 is the largest entry on both factor lists — 48 = 2⁴ · 3 and 60 = 2² · 3 · 5 share 2² · 3 = 12 — and the LCM is 240, because 240 = 2⁴ · 3 · 5 is the smallest number both divide into exactly. Simplifying a fraction wants the GCF; a common denominator wants the LCM; breaking one number down to its primes wants this page. When a problem is really about two numbers, start with the two-number tools: they read the lists this calculator prints, so you never have to read them off by hand.

📚The factor table I look up most

Some numbers come back again and again — in fractions, in timetables, in splitting things into equal groups. These are the ones I keep closest, with their full factor lists and prime factorizations, all worked out by the same trial division the calculator runs:

Notice 120 at the bottom: sixteen factors from just 2³ · 3 · 5, while the prime 97 makes do with two. If a number you need is not in this table, the calculator at the top of the page produces its row instantly — that is, after all, exactly what a factor calculator is for.

📖Factors reference: the numbers people look up most

The table above picks out favourites; this one is the full reference for the numbers searched most — every whole number from 2 to 50, plus 60, 72, 100, 340 and 448. For each: the complete factor list, how many factor pairs it makes, and the prime factorization. Every cell was computed by trial division and independently re-verified before it was printed here.

Pairs counts the factor pairs; the factor count is twice the pair count, except for perfect squares, where the square root pairs with itself and the count is odd.

Two patterns are worth seeing in the data. The primes — 2, 3, 5, 7, 11 and the rest down the table — are exactly the rows with one pair and a prime factorization that is the number itself. And the perfect squares (4, 9, 16, 25, 36, 49 and 100) are the rows where the factor count is odd — 36 has 9 factors and 100 has 9 — because the square root pairs with itself. The heavy rows are the highly composite ones: 48 with 10 factors, 60 with 12, and 340 and 448 out at 12 and 14.

⚠️Mistakes to avoid

• Mixing up factors and multiples. The factors of 12 are 1, 2, 3, 4, 6, 12 (six of them, all ≤ 12). The multiples of 12 are 12, 24, 36, 48 … (infinite, all ≥ 12). Factors divide into a number; multiples are what you get by multiplying out.
• Forgetting 1 and the number itself. Every factor list starts with 1 and ends with n. A “factor count” of 2 means prime — not “no factors”.
• Calling 1 prime. Primes need exactly two factors; 1 has one. It is neither prime nor composite.
• Double-counting the square root. 100 = 10 × 10 gives the pair (10, 10) once — 100 has 9 factors, not 10.
• Assuming bigger means more factors. 97 is prime (2 factors) while the smaller 96 has 12. Factor richness comes from small prime building blocks, not size.