Factor Calculator

What Is a Factor?

Division either comes out even or it does not. A factor is a number for which it does, and the set of factors of any whole number is finite, knowable and short enough to write down — which is what makes factoring the first move in so much of arithmetic.

Factor, Divisor and Multiple: Three Words, Two Directions

Factor and divisor are interchangeable. MathWorld defines a divisor, also called a factor, of a number n as a number d which divides n, so a worksheet asking for the factors of 60 and a textbook asking for its divisors want the same list. Nothing in the mathematics changes with the word.

• Factor or divisor — divides the number exactly. 12 has six of them: 1, 2, 3, 4, 6, 12.
• Proper factor — every factor except the number itself. 12 has five: 1, 2, 3, 4 and 6.
• Common factor — a number that divides two or more numbers. 12 and 18 share 1, 2, 3 and 6, so the greatest common factor is 6.
• Prime factor — a factor that is itself prime. 12's are 2 and 3, and 12 = 2^2 × 3 records how many of each are needed.
• Multiple — the number multiplied by a whole number. 12, 24, 36, 48 and so on, an infinite list rather than a finite one.

Multiple points the other way. The factors of 12 are 1, 2, 3, 4, 6, 12 — six in total, and the list ends at 12. The multiples of 12 are 12, 24, 36, 48 and onward without end. Every factor of a number is less than or equal to it; every multiple is greater than or equal to it; and the statement "3 is a factor of 12" says exactly the same thing as "12 is a multiple of 3".

The word composite covers everything that is neither prime nor 1: a whole number with more than two factors. From 1 to 100 there are 25 primes, 74 composites and one number, 1, that is neither.

Every Factor Has a Partner

Factors are found in pairs, because a division that comes out even produces two whole numbers at once. Dividing 60 by 4 gives 15, so 4 and 15 are both factors and they were discovered by the same operation. Write the pairs out and the list of 12 factors of 60 collapses into six multiplications: 1×60, 2×30, 3×20, 4×15, 5×12, 6×10.

That pairing is why the search can stop at the square root. √60 is a shade under 7.75, and every pair has one member at or below that line and one at or above it — 1 with 60, 2 with 30, 3 with 20, 4 with 15, 5 with 12, 6 with 10. Testing 1 through 7 finds every pair, so no candidate above 7 needs to be tried at all.

Perfect squares are the one case where the pairing doubles back on itself, because their middle pair is a number multiplied by itself. 36 returns 9 factors and the pairs 1×36, 2×18, 3×12, 4×9, 6×6 — the last one uses 6 twice, so the total is odd. That is the whole reason a factor count is odd: between 1 and 100, the numbers with an odd count are exactly 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, the ten perfect squares.

Prime Factors and the One Factorization That Is Unique

A prime is a positive integer greater than 1 with no positive divisors other than 1 and itself, which in this calculator shows up as a factor count of exactly two. The fundamental theorem of arithmetic then says that every positive integer except 1 can be represented in exactly one way, apart from rearrangement, as a product of one or more primes. There is one prime factorization per number, and this tool prints it.

60  =  2 × 2 × 3 × 5
    =  2^2 × 3 × 5

720720  =  2^4 × 3^2 × 5 × 7 × 11 × 13
9999999  =  3^2 × 239 × 4649

The exponent form is a tally, not a shorthand for something looser: 2^2 means the prime 2 appears twice in the multiplication and never a third time. Because the factorization is unique, it is also complete — every factor of 60 is built from the primes 2, 2, 3 and 5, taken with equal or smaller exponents. Choosing none of them gives 1, choosing all of them gives 60, and the other ten choices give the ten factors in between.

This is also the reason 1 is not counted as a prime. MathWorld puts it plainly: if 1 were prime, the statement of the fundamental theorem of arithmetic would have to be modified, since "in exactly one way" would be false because any n = n · 1. Enter 1 here and the panel returns "1 factors" with the factorization written as 1 = 1. Two is the only even prime, and every other prime is odd.

Where a Factor List Actually Gets Used

Factoring is rarely the destination. It is the step that makes the next thing possible, and the same list serves five different jobs:

• Simplifying a fraction. 36/60 needs the largest number dividing both; 36 gives 1, 2, 3, 4, 6, 9, 12, 18, 36 and 60 gives 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60, so 12 is the biggest name on both lists and 36/60 reduces to 3/5.
• Finding a common denominator. The prime factorizations do it: 36 = 2^2 × 3^2 and 60 = 2^2 × 3 × 5, so the least common multiple takes each prime at its highest exponent — 2^2 × 3^2 × 5 = 180.
• Splitting into equal groups. 60 students split evenly only in the 12 ways the factor list allows, and the pairs line names them directly: 1×60, 2×30, 3×20, 4×15, 5×12, 6×10.
• Laying out a rectangle. The same pairs are the whole-number rectangles of area 60 — a 5 by 12 seating plan, a 6 by 10 tile grid, a 4 by 15 garden bed.
• Simplifying a radical. √72 needs the square hiding inside 72, and 72 = 2^3 × 3^2 shows it: 2^3 × 3^2 is 36 × 2, so √72 = 6√2.

Factoring a quadratic leans on the list too. To factor x² + 17x + 60 you need two factors of 60 that add to 17, and reading down the pairs — 1×60, 2×30, 3×20, 4×15, 5×12, 6×10 — the sum 5 + 12 = 17 appears in the fifth pair, giving (x + 5)(x + 12).

Every one of those jobs except the last compares factor lists across two or more numbers rather than reading a single list, and that comparison has its own tool in the GCF Calculator.

How Do You Calculate the Factors of a Number?

Test each candidate divisor in turn and keep the ones that divide evenly. The only refinement that matters is knowing when to stop: the search ends at the square root, because every factor above it was already found as the partner of one below it.

The Factor Formula, Written Out

There are two formulas on this page. The first finds the factors; the second counts them without finding any of them at all, working purely from the prime factorization.

THE FACTOR TEST
for i = 1, 2, 3 … up to √n
    if n mod i = 0   then i and n ÷ i are both factors of n

HOW MANY FACTORS THERE ARE
n = p1^e1 × p2^e2 × … × pr^er
d(n) = (e1 + 1)(e2 + 1) … (er + 1)

"n mod i" is the remainder after dividing n by i, so the test asks whether the division came out even. Each success banks two factors, which is why the loop only has to reach √n: with n = 60 the loop runs i = 1 to 7, succeeds six times, and each success contributes both i and 60 ÷ i for a total of 12. Testing all the way to 60 would run the loop 60 times instead of 7 and would not turn up a single extra factor.

The counting formula is MathWorld's d(n) = ∏(a + 1) over the exponents of the prime factorization, and it is exact rather than an estimate. It works because a factor is a choice of how many copies of each prime to use: 60 = 2^2 × 3 × 5 offers three choices for the 2s (none, one or two), two for the 3 and two for the 5, and 3 × 2 × 2 = 12.

Counting the Factors Before You List Them

Every row below was checked both ways: the exponent product on the left, the factor count this calculator returned on the right.

Two rows carry the lesson of the whole table. 9,999,999 is a seven-digit number with only 12 factors, because 3^2 × 239 × 4649 leans on two large primes that each contribute a mere doubling. 720,720 is smaller and has 240, because its primes are small and there are six different ones. Size has almost nothing to do with it; the shape of the factorization has everything to do with it.

Prime powers are the extreme case in the other direction. A number of the form p^e has exactly e + 1 factors however large it is, which is why 1,024 = 2^10 returns 11 and 9,699,690 — the product of the first eight primes, each used once — returns 256 from eight doublings.

Divisibility Rules That Prune the Search

Working by hand, most candidates can be dismissed without any division at all. These are the rules worth memorizing, in the order they come up:

• 2 — the last digit is even (0, 2, 4, 6 or 8).
• 3 — the digits add to a multiple of 3. For 2,520: 2 + 5 + 2 + 0 = 9.
• 4 — the last two digits form a multiple of 4. For 2,520: 20 ÷ 4 = 5.
• 5 — the last digit is 0 or 5.
• 6 — the rules for 2 and for 3 both hold.
• 8 — the last three digits form a multiple of 8. For 2,520: 520 ÷ 8 = 65.
• 9 — the digits add to a multiple of 9. Again 2 + 5 + 2 + 0 = 9.
• 10 — the last digit is 0.
• 11 — alternately subtract and add the digits; the result is a multiple of 11 (including 0). For 2,520: 2 − 5 + 2 − 0 = −1, so 11 is not a factor.

Applied to 2,520 those rules identify 2, 3, 4, 5, 6, 8, 9 and 10 as factors and rule 11 out, before a single long division. Only 7 needs testing by hand, and 2,520 ÷ 7 = 360. The calculator confirms the shape: 2520 = 2^3 × 3^2 × 5 × 7, with 48 factors in total.

There is no shortcut of this kind for 7, and the rules for larger primes cost more effort than the division they replace. That is the practical boundary of hand factoring — the small primes fall out of the digits, and everything else has to be divided.

Step by Step With This Calculator

One field, one button, four lines of output:

• Type a whole number from 1 to 10,000,000 into the Number box.
• Press Calculate. The page opens with the box empty and the panel showing a dash, so nothing is computed until you do.
• Read the four lines: the factor count as the large readout, then the full list, the prime factorization and the factor pairs.
• Use Reset to clear the box and return the panel to its empty state.

The tool does not recalculate while you type. Changing the number leaves the previous answer on screen until you press Calculate again, which is worth knowing when you are working through a list of numbers and the readout looks suspiciously familiar.

Three input rules cover almost every failed attempt. Thousands separators are rejected — 1,000 returns the amber notice while 1000 works. Decimals are rounded to the nearest whole number rather than refused, so 60.4 and 59.6 are both factored as 60. And leading zeros are harmless: 0060 is read as 60.

Worked Example: The Factors of 60

√60 ≈ 7.746, so the loop tests 1 through 7 and no further. Six of those seven candidates divide 60 evenly, and each one banks its partner at the same moment.

i = 1   60 ÷ 1 = 60   ✓   banks 1 and 60
i = 2   60 ÷ 2 = 30   ✓   banks 2 and 30
i = 3   60 ÷ 3 = 20   ✓   banks 3 and 20
i = 4   60 ÷ 4 = 15   ✓   banks 4 and 15
i = 5   60 ÷ 5 = 12   ✓   banks 5 and 12
i = 6   60 ÷ 6 = 10   ✓   banks 6 and 10
i = 7   remainder 4     ✗

stop: 8 × 8 = 64 is already past 60

Six successful tests, twelve factors, six pairs. The count agrees with the exponent formula on 2^2 × 3 × 5: (2 + 1)(1 + 1)(1 + 1) = 12. The second panel line is the only one with no label in front of it, because the calculator sends the bare list; the other two arrive as a name and a value.

60 earns its place in the worked example. It is the smallest number with 12 factors, and it is the number the hour, the minute and the degree are all still divided on.

Worked Example: A Prime, Where the Search Finds Nothing

97 runs the same loop. √97 ≈ 9.849, so candidates 1 through 9 are tested; 1 succeeds and the other eight fail, which leaves the shortest possible answer.

i = 1   97 ÷ 1 = 97   ✓   banks 1 and 97
i = 2 … 9              ✗   every one leaves a remainder

stop: 10 × 10 = 100 is already past 97

A count of two is the calculator's primality test, and the factorization line adds the sentence in full rather than leaving you to infer it. The same thing happens at the top of the range: 9,999,991 returns "2 factors" and the list 1, 9999991, after testing every candidate up to 3,162 and finding not one of them divides.

Every claim of the form "7 is a factor of 2,520" is one division away from being checked, remainder and all, in the Division Calculator.

Trial division is slow in theory and instant here. Factoring the largest number this tool accepts takes well under a millisecond, because the loop never runs past √10,000,000 — 3,162 iterations at the very most, whatever the number turns out to be.

Factor Chart: Every Number From 1 to 100

The complete reference, generated by running this calculator on each number in turn. The "How many" column is the tool's large readout, and the factorization column is what it prints after the equals sign, with a caret for each exponent.

Factors of 1 to 50

Rows with a count of 2 are the primes, and for those the panel adds "— it's prime (only 1 and itself divide it)" to the factorization line. Row 1 is the only number in the table that is neither prime nor composite: it has one factor, itself.

Factors of 51 to 100

The second half of the chart holds ten primes against fifteen in the first half. Primes thin out as numbers grow, but they do it slowly and unevenly — 89 and 97 are both here, with seven composites between them.

Four Patterns in the Chart

A chart this size is worth reading for its shape as well as for individual lookups. Four things fall out of it:

• Exactly 25 of the first hundred numbers are prime, each with a count of 2: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
• Exactly 10 have an odd count, and they are precisely the perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
• The most common count is 4, shared by 32 numbers — every product of two distinct primes, such as 6, 10, 14, 15, 21 and 22, plus the cubes 8 and 27.
• The record for this range is 12 factors, held jointly by five numbers: 60, 72, 84, 90, 96.

The record-holders are worth comparing, because they reach the same total by different routes: 60 = 2^2 × 3 × 5, 72 = 2^3 × 3^2, 84 = 2^2 × 3 × 7, 90 = 2 × 3^2 × 5, 96 = 2^5 × 3. Three small primes with one of them squared, a cube times a square, or a fifth power times a prime — the exponent product lands on 12 every time.

Two rows in the chart are quietly famous. 6 has factors 1, 2, 3, 6, and its factors below itself add to 1 + 2 + 3 = 6; 28 has 1, 2, 4, 7, 14, 28, and 1 + 2 + 4 + 7 + 14 = 28. Numbers that equal the sum of their own smaller factors are called perfect, and they are scarce: after 6 and 28 the next two are 496, which this calculator returns 10 factors for, and 8,128, with 14.

The gaps matter as much as the entries. A count of 11 appears nowhere in the first hundred numbers, because 11 is prime and the only way to reach it is p^10 — the smallest of which is 1,024, ten times past the end of the chart.

Factor Examples: The Numbers People Look Up

Past 100 the lookups cluster: round numbers, powers, and the handful of numbers that carry an unusual quantity of factors. Every row here is this calculator's own output.

Factors of Common Numbers Above 100

Numbers ending in a run of zeros are less generous than they look. 500 returns 12 factors while 480, a smaller number, returns 24, because 500 = 2^2 × 5^3 has only two distinct primes while 480 = 2^5 × 3 × 5 has three.

Factors of Larger Round Numbers

The 1,000 row is a useful one to have in full: 16 factors, 2^3 × 5^3, and the pairs 1×1000, 2×500, 4×250, 5×200, 8×125, 10×100, 20×50, 25×40 — eight pairs exactly, which is the most the pairs line prints, so every one of them shows with no ellipsis.

576 and 900 both return odd counts — 21 and 27 — which by the rule above means both are perfect squares, at 24² and 30².

Powers of a Single Prime

When a number is a prime raised to a power, the factor list is just that prime's own powers, and the count is the exponent plus one. Nothing else is possible, because there is no second prime to combine with.

65,536 is 2^16 and returns 17 factors. 720 is more than ninety times smaller and returns 30. A long factor list needs several different primes, not a great many copies of one.

Highly Composite Numbers: Where the Factors Pile Up

A highly composite number has more divisors than every smaller number does. MathWorld gives the start of the sequence as 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, and running each one through this calculator returns 1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 32, 36, 40, 48, 60 factors respectively — each total larger than any that came before it.

Read down the factorization column and the recipe is obvious: small primes, in descending exponents, with no gaps. 2^4 × 3^2 × 5 × 7 uses 2, 3, 5 and 7 — the four smallest primes — and skips none of them. A number the same size built from larger primes cannot compete: 5,040 returns 60 factors while 5,041, which is 71², returns 3.

These are the numbers that end up in units of measure and in packaging. 12, 24, 60, 360 and 720 all sit on the list, and 2,520 is the smallest number divisible by every one of 1 through 10 — its factorization, 2^3 × 3^2 × 5 × 7, contains exactly what those ten numbers need between them.

That last property of 2,520 is a least common multiple rather than a factor question, and going the other way — from several numbers to the smallest one they all divide into — is the job of the LCM Calculator.

Factor Pairs, Which Are the Equal-Group Splits

The fourth panel line reads as a list of multiplications, and it answers the practical question directly: in how many ways can this quantity be split evenly, and into what sizes?

A pair is a rectangle as much as it is a division. 24 offers 1×24, 2×12, 3×8, 4×6, so a class of 24 makes four equal rows of 6 or six rows of 4, and a tray of 24 tiles lays out as 3 by 8 or 4 by 6.

The last two rows end in an ellipsis because the line stops after eight pairs. 180 has 18 factors and therefore nine pairs, one more than the panel prints; 240 has 20 and ten. The full list on the second line still contains every factor.

How to Read Your Result

There are no bands or categories here — a factor list is not a score. Reading the result is a matter of knowing what each of the four lines is for, what the count on its own already tells you, and where the display stops short of the full answer.

The Four Lines the Panel Returns

Entering 24 and pressing Calculate produces these, in order:

• The large readout under the heading Factors Found: "8 factors". It is a count, not a factor — how many, not which.
• The list itself, printed plain with no label in front of it: 1, 2, 3, 4, 6, 8, 12, 24. Always in ascending order, always starting at 1 and ending at the number itself.
• Prime factorization: 24 = 2^3 × 3. The caret means exponent, so 2^3 is three 2s multiplied together.
• Factor pairs: 1×24, 2×12, 3×8, 4×6. Each pair multiplies back to your number.

The Copy result button below the panel puts the whole result on the clipboard as text, which beats selecting a fifty-entry list by hand.

Line one and line two answer different questions, and they are worth keeping apart. "8 factors" does not mean 8 is a factor of 24, although it happens to be; it means the list on the next line has eight entries in it.

What the Count Alone Tells You

Before reading the list, the count has already settled several questions:

The primality shortcut is the most useful of these. Any number this calculator accepts can be tested by entering it and glancing at the readout: 9,973 returns "2 factors" and 9,999,991 returns "2 factors", so both are prime, while 9,999,999 returns "12 factors" and is emphatically not.

• 1 factor — the number is 1. It is the only number with a single factor, and the panel prints "1 factors", plural and all.
• 2 factors — the number is prime. The factorization line says so outright: "97 = 97 — it's prime (only 1 and itself divide it)".
• 3 factors — the number is the square of a prime. 49 returns 1, 7, 49 and 49 = 7^2; the same holds for 4, 9, 25, 121 and 169.
• An odd count — the number is a perfect square, because its middle pair is a number times itself. 144 returns 15 factors, and 12 × 12 sits at the end of its pairs line.
• An even count — the number is not a perfect square, and every factor has a distinct partner.
• A large count — the number is built from several small primes. 720,720 returns 240 factors from 2^4 × 3^2 × 5 × 7 × 11 × 13.

When the Display Stops Short

Two lines are capped, and both say so where they stop. The list prints the first 48 factors and then adds a count of the rest, while the pairs line prints eight pairs and ends in a bare ellipsis. The third column below is the literal end of the list line: where it carries a "(+N more)", the display was cut.

2,520 is the boundary case for the list: with exactly 48 factors it prints in full, and 5,040 with 60 does not, ending "336… (+12 more)" instead. For the pairs line the boundary is 16 factors — anything with 17 or more loses the tail.

Nothing is lost from the calculation itself, only from the display. The count on the first line is always the true total, and the factorization on the third line is always complete, so a truncated list can be reconstructed from the primes if you need every entry.

The One Message That Replaces an Answer

A single amber notice covers every rejected input: "Check your inputs" above the line "Enter a whole number from 1 to 10,000,000". It appears for six different reasons:

• The box is empty, or holds something that is not a number.
• The value is 0, which every whole number divides — the list would never end.
• The value is negative.
• The value is above 10,000,000. Exactly 10,000,000 is accepted and returns 64 factors.
• The value carries a thousands separator, as in 1,000. Type 1000.
• The value is a decimal below 0.5, such as 0.4, which rounds to 0 and is then rejected.

Rounding is the quiet one, because it does not produce a message. A decimal from 0.5 upward is rounded and factored without comment: 0.5 becomes 1, 59.6 and 60.4 both become 60, and 60.5 becomes 61 — which returns "2 factors" rather than the twelve you were expecting from 60. If a result looks nothing like the number you typed, check whether it had a decimal point in it.

Limits: When This Calculator Does Not Apply

The tool does one job: the positive whole-number factors of one positive whole number at a time. These are the edges of that job.

• One number at a time. There is no second box, so a greatest common factor or a least common multiple has to be assembled by running each number separately and comparing the lists.
• Whole numbers from 1 to 10,000,000 only. Anything larger returns the amber notice, and there is no way to raise the ceiling from the page.
• Decimals are rounded, not refused, and no warning is shown. 60.5 is factored as 61 and returns a completely different answer from 60.
• Negative numbers are rejected. Among the integers −3 divides 12 exactly as 3 does; this calculator lists positive factors only, which is the school convention.
• Zero is rejected. Every whole number divides 0 exactly, so the honest answer is an infinite list and the tool declines to start one.
• The number 1 returns "1 factors", with the plural left in and the factorization printed as 1 = 1. 1 is neither prime nor composite.
• Exponents are printed with a caret, not as superscripts: 144 = 2^4 × 3^2 rather than 2⁴ × 3². The meaning is identical; the typography is not.
• The factor list stops after 48 entries and the pairs line after 8, so the largest results are shown in part rather than in full.
• It factors numbers, not expressions. There is nothing here for x² + 5x + 6, and nothing for polynomials, fractions or algebraic terms.
• It finds factors, not their sums or products. Perfect, abundant and deficient numbers have to be judged by adding the list up yourself.
• The method is trial division, which is why it is capped. It is instant to 10 million and hopeless at the hundreds of digits used in public-key cryptography, where the difficulty of factoring is the point rather than an obstacle.

The expression case is worth naming on its own. Factoring x² + 17x + 60 into (x + 5)(x + 12) starts with the factor pairs of 60, but finishing it — or handling a quadratic whose roots are not whole numbers at all — belongs to the Quadratic Equation Solver.

One thing that looks like a limit is not. A large number is neither a slow one nor a rich one: 9,999,991 is prime, settling in under a millisecond with just two factors, while 720,720 — almost fourteen times smaller — carries 240.