Factor calculator, factors and prime factors.

A factor calculator lists every whole number that divides a given number exactly. The factors of 12 are 1, 2, 3, 4, 6 and 12. This page also splits the number into primes, which is its prime factorisation, pairs the factors up, counts them, adds them, and draws a factor tree. Everything is exact, so it works for numbers far bigger than a pocket calculator can hold.

How to use the factor calculator

1. Type a whole number in Number. The answer appears as you type.
2. Read the big card for the prime factorisation, or the message that the number is prime. The smaller cards give the number of factors, their sum, the type and whether it is a square or a cube.
3. Open Factor pairs and All factors for the lists, and Steps and Factor tree for the working.
4. Use − 1 and + 1 to step to the next number, Random for a number to try, and Include negative factors if your course counts them.
5. Use Copy result to copy the answer, or Copy link to share the number.

A worked example: 360

Divide by the smallest prime that fits, again and again. 360 ÷ 2 = 180, 180 ÷ 2 = 90, 90 ÷ 2 = 45, then 45 ÷ 3 = 15, 15 ÷ 3 = 5, and 5 is prime. So 360 = 2³ × 3² × 5. To count the factors, add 1 to each exponent and multiply: (3 + 1) × (2 + 1) × (1 + 1) = 24. To add them, multiply the sums for each prime: (1 + 2 + 4 + 8) × (1 + 3 + 9) × (1 + 5) = 15 × 13 × 6 = 1170.

The 24 factors make 12 factor pairs: 1 × 360, 2 × 180, 3 × 120, 4 × 90, 5 × 72, 6 × 60, 8 × 45, 9 × 40, 10 × 36, 12 × 30, 15 × 24 and 18 × 20. The pairs stop at the square root, since after that the same pairs repeat in reverse. The square root of 360 is a little under 19, and the last pair is 18 × 20.

Prime or composite

A prime has exactly two factors, 1 and itself, as 97 does. A composite number has more, as 360 does. The number 1 is neither: it has only one factor. A quick check by hand is to try the primes up to the square root. For 97 that means 2, 3, 5 and 7, and none of them divide it, so it is prime.

Perfect, abundant and deficient numbers

Add up the proper factors, which are all the factors except the number itself, and compare the total with the number. If they are equal, the number is perfect. If the total is bigger, it is abundant. If smaller, it is deficient.

Every prime is deficient, because its only proper factor is 1. The card named Type shows the answer for your number.

Squares and cubes

A number is a square when every exponent in its prime factorisation is even, and a cube when every exponent is a multiple of 3. For example 36 = 2² × 3² is a square, 6 × 6, and 27 = 3³ is a cube. Squares are also the numbers with an odd count of factors, because one pair is made of the same number twice.

Big numbers and the time limit

Numbers up to about a trillion are split by trial division with a wheel that skips multiples of 2, 3 and 5. Bigger ones use a primality test and Pollard's rho method. Both run inside your browser for a couple of seconds at most. The number 600851475143 splits at once into 71 × 839 × 1471 × 6857. A number made of two very large primes, of about 12 digits each or more, can take longer than the limit. In that case the page shows the factors it found, marks what is left as composite, and says it did not finish. It never guesses.

Common problems

• Decimals such as 2.5 have no factors in this sense. Use whole numbers.
• 0 is not listed, because every number divides it.
• A minus sign is ignored for the main answer. Tick Include negative factors to list the negative ones too.

To find the greatest common factor or least common multiple of several numbers, use the GCF and LCM calculator. For the remainder after a division, try the remainder calculator, and to simplify a fraction, the fraction calculator.