What a factor of a number is
A factor of a whole number is any whole number that divides it exactly, with nothing left over. 36 is divisible by 1, 2, 3, 4, 6, 9, 12, 18 and 36 itself, so those nine numbers are its factors; try 5 or 7 and a remainder is left behind, so neither qualifies. Some textbooks call the same idea a divisor; the two words mean the same thing, and this page uses them interchangeably.
Every factor comes with a partner. Divide 36 by 4 and the answer is 9, so 4 and 9 are a matching pair: 4 × 9 = 36. Every factor has exactly one such partner, the idea behind the factor pairs table further down this page.
Finding every factor by trial division
The plainest way to find every factor is to try dividing the number by 1, then 2, then 3, and so on, keeping whichever whole numbers divide it exactly. That works, but every factor found below the square root hands over its partner above the square root for free: once 4 is confirmed as a factor of 36, its partner 9 is already known too, from 36 ÷ 4, so there is no need to separately check whether 9 divides in.
That is why this calculator only checks whole numbers up to √n (n's square root), rounded down, rather than all the way up to n. Past that point, every division would just rediscover a pair already found the other way round. For 36, √36 is exactly 6, so checking stops at 6; for a number without a whole-number square root, checking stops at the largest whole number below it. This one idea keeps factorising fast: a trillion only needs checking up to about a million.
When the square root itself divides the number exactly, as 6 does for 36, that factor pairs with itself and is only counted once, which is also why a perfect square ends up with an odd number of factors overall, covered further down.
Prime factorisation and the exponent form
Every whole number above 1 breaks down into a unique product of prime numbers, in exactly one way, ignoring the order they are multiplied in. This is sometimes called the fundamental theorem of arithmetic. 36 breaks down into 2 × 2 × 3 × 3, written more compactly with exponents as 2² × 3², meaning the prime 2 appears twice and the prime 3 appears twice.
This calculator finds that breakdown with the same trial-division idea as above: divide out 2 for as long as it divides exactly, counting how many times it does, then move on to 3, then 5, and so on, stopping at √n for the same reason as before. Whatever is left over once no more primes up to √n divide in is itself a prime, and gets one final exponent of 1. A number with nothing left to divide out below its own square root, like 97, is a prime number outright.
The shortcut for counting factors, without listing a single one
Once a number's prime factorisation is known, its total number of factors can be worked out directly, without writing out a single factor. Add 1 to each exponent in the factorisation, then multiply those results together.
For 36 = 2² × 3², both exponents are 2, so the count is (2 + 1) × (2 + 1) = 3 × 3 = 9, matching the nine factors listed above exactly. It works because every factor of 36 is built by choosing how many 2s to include (0, 1 or 2, three choices) and separately how many 3s to include (also three choices); every combination multiplies out to a different, valid factor, giving 9 in total. A prime such as 97 has the exponent 1 on a single prime, giving (1 + 1) = 2 factors: 1 and 97.
Prime, composite, and the special case of 1
A prime number has exactly two factors: 1 and itself. A composite number has more than two. 1 sits outside both definitions, since it has only one factor (itself), fitting neither the "exactly two" rule for prime nor the "more than two" rule for composite; mathematicians class it as neither, a category of its own often called a unit.
Counting factors gives an instant prime-or-composite check: exactly two means prime, more than two means composite, exactly one (only ever the number 1) means neither. There is no need to hunt for a factor that "proves" a number composite; the count from the pairs table above already settles it.
Perfect, abundant and deficient numbers
A number's proper factors are every factor except itself; 36's proper factors are 1, 2, 3, 4, 6, 9, 12 and 18. Adding those up and comparing the total against the original number sorts every whole number above 1 into one of three classes.
6 is the smallest perfect number: 1 + 2 + 3 = 6 exactly. 28 is the next: 1 + 2 + 4 + 7 + 14 = 28. Perfect numbers are strikingly rare; only four sit below ten thousand (6, 28, 496 and 8,128), and every even one found so far follows a pattern Euclid described over two thousand years ago.
12 is the smallest abundant number: its proper factors 1, 2, 3, 4 and 6 add up to 16, four more than 12. Abundant numbers tend to have several small factors packed close together. Deficient numbers are the ordinary case: every prime is deficient, since its only proper factor is 1, always less than itself, and most composites turn out deficient too.
Why only perfect squares have an odd number of factors
Factors normally come in pairs, d and n ÷ d, and every pair adds two to the running total, so the count stays even. 28's factors, 1, 2, 4, 7, 14 and 28, form three complete pairs, giving an even count of 6. A perfect square breaks that pattern in one place: its square root pairs with itself, contributing only one factor rather than two, tipping the count from even to odd. 36's square root, 6, is exactly this case: 6 × 6 = 36, so 6 is counted once, leaving 36 with 9 factors.
The logic runs both ways: a number has an odd number of factors if, and only if, it is a perfect square. Nothing else can produce an odd count, since every other factor is guaranteed a distinct partner.
Limits, zero, and messy input
This calculator checks factors by trial division up to √n, and states its limit honestly: numbers up to 1,000,000,000,000 (one trillion) are handled instantly and exactly, since the square root of a trillion is only a million, a search a browser finishes without any noticeable delay. Above that limit, the page stops rather than churn through a search that would start to feel sluggish, and says so plainly instead of guessing.
0 is a special case rather than an error: every whole number divides 0 exactly, so 0 has infinitely many factors rather than a finite list, and this page says so instead of enumerating an endless one. 1 is the other edge case: its only factor is itself, so the pairs table shows just one row, and 1 carries none of the prime, composite, perfect or abundant labels that apply to every other whole number.
The number box reads pasted input generously: thousands commas, a leading currency symbol, or surrounding brackets are all stripped away automatically. A decimal point keeps only the whole-number part in front of it, and a negative sign is dropped, since factors are only discussed for 0 and the positive whole numbers; either adjustment is reported back in a note rather than silently applied.
How this differs from the greatest common factor calculator
This page answers a question about one number on its own: every divisor it has, paired up, prime-factorised and classified. The Greatest Common Factor Calculator answers a different question, about two or more numbers together: the largest factor they all share. Both tools use trial division and prime factorisation, just pointed at different problems.
Questions people ask
What is the difference between a factor and a multiple?
A factor of n divides into n exactly and is always less than or equal to n (for positive n). A multiple of n is what you get by multiplying n by a whole number, and is always greater than or equal to n. 4 is a factor of 12; 48 is a multiple of 12. The two ideas run in opposite directions.
Why does checking stop at the square root instead of going all the way to n?
Because every factor above the square root already turned up as the partner of a factor found below it. Once 4 is confirmed as a factor of 36, its partner 9 is already known from 36 ÷ 4, so separately checking whether 9 divides 36 would only repeat information already found.
Is 1 a prime number?
No. A prime number needs exactly two distinct factors, 1 and itself. 1 has only one factor, itself, so it fits neither the prime definition nor the composite one (which needs more than two factors); it sits in a category of its own.
What does it mean for a number to be "abundant"?
It means the number's proper factors (every factor except itself) add up to more than the number itself. 12 is the smallest example: 1, 2, 3, 4 and 6 add up to 16, four more than 12. Most whole numbers are not abundant; the deficient case, where the proper factors add up to less, is far more common.
Why do perfect squares always have an odd number of factors?
Because their square root pairs with itself rather than with a different partner. Every other factor pairs off with a distinct partner, adding two to the count; a perfect square's square root only adds one, which tips the total from even to odd.
Is there a limit to how large a number this calculator can factorise?
Yes, stated plainly: numbers up to 1,000,000,000,000 (one trillion) are factorised instantly by trial division up to their square root. Above that limit, the search would need too many steps to stay quick in a browser, so the page stops and says so rather than freezing or guessing.
Every result on this page comes from exact whole-number trial division; nothing here is rounded or approximated at any step, up to the stated limit.