Factor Calculator

60 ÷ 1 = 60 ⇒ 1 * 60; 60 ÷ 2 = 30 ⇒ 2 * 30; 60 ÷ 3 = 20 ⇒ 3 * 20; 60 ÷ 4 = 15 ⇒ 4 * 15; 60 ÷ 5 = 12 ⇒ 5 * 12; 60 ÷ 6 = 10 ⇒ 6 * 10; 60 ÷ 7 is not exact, 8 × 8 > 60, stop

n

The number being factored — a whole number from 1 to 1000000. One is allowed and gives a single factor, 1, which is a real answer rather than a degenerate one: 1 is its own factor and divides nothing else evenly except the numbers that are multiples of it

d

A trial divisor, tested in order 1, 2, 3, 4 and so on. The page never tests past the square root of n, and that cutoff is the whole trick: if d × e = n and d is bigger than the square root, then e is smaller than it, so e was already found and the pair was already printed

d * d ≤ n

The stopping rule, written out. For 60 the last divisor tested is 7, because 7 × 7 = 49 is under 60 and 8 × 8 = 64 is over it. The rule is why the number of trials grows with the square root of the input rather than with the input — a million takes a thousand trials, not a million

n / d

The partner of a factor. When d divides n exactly, the quotient is the other half of the pair, and the page produces it in the same step. This is why 60's pairs come out as 1 * 60, 2 * 30, 3 * 20 and so on in order, with no sorting: the smaller halves are found in ascending order, so the pairs are too

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Every factor of 60 as one flat list, ascending. Twelve entries, which is an even count — the pairing is complete, every factor has a partner. The list is the same information as the pairs below it, rearranged: read the pairs and collect the numbers, and you have this line back

6 * 6

The middle pair of 36, and the only shape in which a factor appears twice on one line. 36 is a perfect square, so its factor count is odd — 1, 2, 3, 4, 6, 9, 12, 18, 36, which is nine — and the page prints the middle pair once rather than twice. This is why a perfect square never has a factor count that is even

Simplifying a fraction is the most common reason to want a factor list. To reduce 42/60 by hand you look for a number dividing both, and the factors of 60 tell you the candidates while the factors of 42 tell you which ones qualify — the largest such number is the greatest common factor, and gcf-calculator answers that question directly if that is all you need. A second use is a rectangle with a fixed area: a garden bed of 60 square metres can be 1 by 60, 2 by 30, 3 by 20, 4 by 15, 5 by 12 or 6 by 10, and the factor pairs are exactly that list of possible shapes, which is why they are printed in the shape they are. The same question appears whenever items have to be arranged in equal rows and columns: seating 60 guests at tables of equal size, or splitting a class into groups of the same size with nobody left over. A third use is checking a claim. If someone says 91 is prime, its factor list settles it — 1, 7, 13, 91 — and the fact that there are four entries rather than two is the refutation. Divisibility rules for 2, 3, 5 and 9 cover the small cases by hand, and this page picks up from where those stop being easy. When the question is really about the pieces rather than the factors, the answer lives elsewhere: prime-factors-calculator breaks the number into primes, which this page deliberately does not do; lcm-calculator and gcf-calculator compare two numbers rather than describing one; and simplify-fractions-calculator uses a common factor to reduce a fraction directly, without showing you the list.

1. Twelve factors: 60

   1. Test 1: 60 ÷ 1 = 60 exactly, so 1 and 60 are both factors
   2. Test 2: 60 ÷ 2 = 30, so 2 and 30 are factors; test 3: 60 ÷ 3 = 20, giving 3 and 20
   3. Test 4: 60 ÷ 4 = 15, giving 4 and 15; test 5: 60 ÷ 5 = 12, giving 5 and 12
   4. Test 6: 60 ÷ 6 = 10, giving 6 and 10; test 7: 60 ÷ 7 is not a whole number
   5. 8 × 8 = 64 is past 60, so stop — the twelve factors are complete

   The default input, and a good one because 60 has many factors — twelve, the most of any number below it. Note how cheap the search was: seven trial divisors, not sixty. The pairs line and the factor list are the same twelve numbers; the pairs are simply folded so that each one sits next to the number it multiplies with. Also worth noticing that the pair list ends at 6 * 10 rather than continuing into 10 * 6: once the smaller half of a pair passes the square root, every remaining pair is a repeat of one already printed.

2. A perfect square: 36

   1. 1 and 36 pair, then 2 and 18, then 3 and 12, then 4 and 9
   2. 6 × 6 = 36, so 6 is its own partner — the middle of the list
   3. 7 does not divide 36; 7 × 7 = 49 is already past it, but 6 was the last divisor under the square root
   4. The factors are 1, 2, 3, 4, 6, 9, 12, 18, 36 — nine of them, an odd count

   The case that shows why a factor count can be odd. Every factor except the square root has a distinct partner, so they pair off and contribute an even number; the square root pairs with itself and contributes one. That is the whole reason 36 has nine factors while 60, a larger number, has twelve, and 12 has six. The pairs line prints 6 * 6 once rather than twice, which is a choice — printing it twice would make the two lines disagree about how many factors there are.

3. A prime number: 7

   1. Test 1: 7 ÷ 1 = 7 exactly, so 1 and 7 are factors
   2. Test 2: 7 ÷ 2 is not a whole number; 3 × 3 = 9 is past 7, so the search stops there
   3. Nothing between 1 and 7 divides it
   4. Two factors and one pair — which is the definition of a prime number

   The shortest non-trivial output on the page, and the one that shows a prime is not a failure to factor. A prime is a number with exactly two factors, and those two are always 1 and the number itself; the page reports that fact rather than reporting that it could not find anything. The stopping rule does real work here — after testing 2 there is nothing left to test, because 3 × 3 is already greater than 7. Testing 3, 4, 5 and 6 would cost four divisions and could not possibly find anything.

The input must be a whole number from 1 to 1000000. Zero is refused: every whole number divides zero, so its factor list would be infinite and there is nothing sensible to print. Negative numbers are refused too, even though ±1, ±2 and so on all divide −60 — the page reports the positive factors of a positive number, and a signed factor list would need a convention about whether both signs are shown that this page does not state. Decimals are refused rather than rounded. The ceiling of one million is what keeps the trial division bounded: the search stops at the square root, so the worst case is a thousand trials, and the answer list can be long but never unmanageable. It can still be long — 720720 has 240 factors, the most of any number below one million — and a list that size is correct but not much fun to read. The page reports factors and factor pairs, not prime factors, and the difference is easy to miss: 60's factor list contains 2, 3 and 5, but the page never says 60 = 2² × 3 × 5, and for a large number the two questions have very different costs. Divisibility rules for 2, 3, 5 and 9 are not called out separately — they are simply absorbed into the trial division, so a number ending in 0 shows 2, 5 and 10 as factors without the page mentioning why. Every factor is printed as a whole number in a comma-separated list, and the pairs use an asterisk for multiplication; no thousands separators are used, so a factor of one million prints as 1000000. Finally, the reference table below shows three fixed numbers rather than following your input; the panel above is the one that answers what you typed.

What exactly counts as a factor?

Any whole number that divides your number exactly, leaving no remainder. Both members of each pair count, so for 60 the list runs from 1 all the way up to 60 — 1 and the number itself are always factors, and the interesting ones are in between. Fractions do not count even when they divide evenly: 2.5 goes into 60 twenty-four times, but 2.5 is not a whole number, so it is not a factor. This page lists the positive factors of a positive number, which is the standard convention.

Why does the search stop at the square root?

Because past it, every factor you would find is the partner of one you have already seen. If d × e = 60 and d is larger than the square root of 60, then e is smaller than it — so e was tested earlier and the pair was printed then. Testing beyond that point can only rediscover pairs in the other order. This is why 60 takes seven trial divisors rather than sixty, and why the cost of this page grows with the square root of the input: a number near a million takes about a thousand tests, not a million.

Why does 36 have an odd number of factors?

Because 36 is a perfect square, so its square root pairs with itself. Every other factor has a different partner — 1 with 36, 2 with 18, 3 with 12, 4 with 9 — and each such pair adds two to the count. Then 6 × 6 = 36 adds one. So the total is 4 × 2 + 1 = 9. In general a number has an odd number of factors exactly when it is a perfect square, and that is the quickest way to tell from a factor list whether the number is one.

Is this the same as prime factorization?

No, and it is worth being clear about the difference. This page gives every factor; prime factorization breaks the number into primes multiplied together. For 60 the prime factorization is 2² × 3 × 5, and the page never prints that — though 2, 3 and 5 all appear in the factor list, because primes are factors like any others. The practical difference is cost: this page does work proportional to the square root of the number, while prime factorization is genuinely hard for large numbers, which is the assumption public-key cryptography rests on.

Can I enter 0 or a negative number?

No, and neither is an arbitrary restriction. Every whole number divides 0, so its factor list would never end. For a negative number the factors come in both signs — 1 and −1, 2 and −2, and so on — and this page would have to take a position on whether to list both, which it does not. The input is a whole number from 1 to 1000000. Decimals are refused as well rather than rounded, because rounding would silently answer a question about a different number.

Why are the factors split into two columns?

Because they answer two different questions. The flat list is what you check when you want to know whether some number divides yours. The pairs are what you want when the number is a product of two things you are choosing — a rectangle with that area, a set of equal rows and columns, a group split with nobody left over. A 60-square-metre bed can be 1 by 60, 2 by 30, 3 by 20, 4 by 15, 5 by 12 or 6 by 10, and those six shapes are exactly the pairs column. The two columns hold the same numbers; only the arrangement differs.

• Divisor — what it means for one whole number to divide another exactly, and the convention that every number divides itself — Wolfram MathWorld (United States)
• Prime Number — numbers with exactly two factors, and why that definition is the one the factor list makes visible — Wolfram MathWorld (United States)
• Perfect Square — numbers whose square root is a whole number, and the reason their factor count is always odd — Wolfram MathWorld (United States)
• 教育部关于印发义务教育课程方案和课程标准（2022 年版）的通知——The fifth item in the annex list of this notice is the Mathematics Curriculum Standards for Compulsory Education (2022 edition); factors and multiples are part of the Number and Algebra strand in the upper primary grades, and the wording of the standards and the grade-band breakdown are governed by that annex — 中华人民共和国教育部