Factor Calculator

What this calculator does

So, you want every factor of a whole number, the complete list of numbers that divide into it evenly. This tool finds them all. Enter a positive whole number and it returns each of its factors, from 1 up to the number itself.

One input, one button, and a full list of factors comes back.

How to use it

1. Enter a positive whole number.
2. Press Calculate.

What a factor is

A factor of a number is any whole number that divides it exactly, leaving no remainder. The factors of 12 are 1, 2, 3, 4, 6, and 12, because each of those divides 12 with nothing left over. Two numbers are missing from that list, 5 and anything above 6 apart from 12, because they do not divide 12 evenly. Every number has at least two factors, 1 and itself, since any number can be divided by 1 and by its own value. The tool checks each whole number up to your input and keeps the ones that divide it cleanly.

Factors come in pairs

There is a pattern hiding in any factor list: factors come in pairs that multiply to give the number. For 12, the pairs are 1 and 12, 2 and 6, and 3 and 4. Each pair multiplies back to 12. This pairing is why you never have to search all the way up to the number to find its factors: once you pass its square root, every remaining factor is just the partner of one you have already found. It also explains why perfect squares have an odd number of factors, since one pair is a number multiplied by itself.

All factors, not just the prime ones

It is worth being clear about what this tool lists, since factors come in different flavours. This tool gives every factor, prime or not. For 12 that means 1, 2, 3, 4, 6, and 12, including 4 and 6, which are not prime, and 1, which is neither prime nor composite. That is different from the prime factors of a number, which are only the prime building blocks. If you want just those, the prime factor calculator lists the distinct primes, and the prime factorization calculator writes the number as a product of primes. This tool casts the wider net and returns the whole set of divisors.

A worked example

Enter 12. The tool checks each number from 1 to 12 and keeps those that divide evenly: 1, 2, 3, 4, 6, and 12. It skips 5, 7, 8, 9, 10, and 11, none of which divide 12 without a remainder. So the factors of 12 are 1, 2, 3, 4, 6, 12, which fall into the pairs 1 and 12, 2 and 6, and 3 and 4.

Where listing factors helps

Knowing all the factors of a number is useful in plenty of everyday situations. It tells you the ways you can arrange things into equal rows and columns, since the factors are exactly the group sizes that divide evenly. It underpins simplifying fractions and finding common factors between numbers. And it is the quickest way to see whether a number is prime: a prime has exactly two factors, 1 and itself, so if the list is longer than that, the number is composite.

Questions people ask

What is a factor of a number?

A whole number that divides it evenly, with no remainder. The factors of 12 are 1, 2, 3, 4, 6, and 12.

Why do factors come in pairs?

Because each factor multiplies with another to give the number. For 12 the pairs are 1 and 12, 2 and 6, and 3 and 4.

Are 1 and the number itself factors?

Yes. Every number is divisible by 1 and by itself, so both always appear in the list.

Is this the same as prime factors?

No. This lists every factor, including non-prime ones like 4 and 6. For only the primes, use the prime factor calculator.

How does this show if a number is prime?

A prime has exactly two factors, 1 and itself. If the list has more than two, the number is composite.

References

On factors and divisors. A factor of a number is a whole number that divides it without remainder, and the factors include 1 and the number itself.

1. Eric W. Weisstein, "Greatest Common Divisor," from MathWorld, a Wolfram resource, on divisors, the numbers that divide another exactly.
2. Eric W. Weisstein, "Prime Factor," from MathWorld, a Wolfram resource, on the prime factors, a subset of all the factors of a number.