Prime Factorization Calculator

1. Type the whole number you want to factor. Any number from 2 to 999,999,999,999 works.
2. Press Calculate, or just press Enter. The answer appears first, followed by the supporting numbers and the step-by-step division.
3. Read the exponent form first, then the expanded form, the factor counts, and the divisor count below it.
4. Press Reset to clear the form and try another number.

Prime factorization breaks a whole number into the prime numbers that multiply together to make it. A prime is a whole number greater than 1 whose only divisors are 1 and itself. The primes start 2, 3, 5, 7, 11, and continue without end. The number 2 is the only even prime, because every larger even number is divisible by 2. The prime factorization of 60 is 2^2 x 3 x 5, because 2 x 2 x 3 x 5 = 60. Every whole number greater than 1 has exactly one prime factorization, apart from the order of the factors, so 60 always breaks into the same primes no matter which method you use.

The calculator shows the same answer three ways.

Exponent Form vs Expanded Form vs CSV

Exponent form groups repeated primes as powers. For 360 the answer is 2^3 x 3^2 x 5. The exponent counts how many times each prime appears, so 2^3 means 2 x 2 x 2 = 8 and 3^2 means 3 x 3 = 9, which gives 8 x 9 x 5 = 360. Expanded form lists every prime on its own, so 360 becomes 2 x 2 x 2 x 3 x 3 x 5. The CSV form is the same list with commas, 2, 2, 2, 3, 3, 5, which is handy for pasting into a spreadsheet. Under the forms you also get the distinct prime factors, the total count with multiplicity, whether the number is prime, and how many divisors the number has.

A factor tree splits a number into any two factors, then splits each composite factor again, and keeps going until only primes remain. For 60, split 60 into 6 x 10. Then split 6 into 2 x 3 and 10 into 2 x 5. The primes at the ends are 2, 2, 3, and 5, so 60 = 2^2 x 3 x 5. Any valid tree gives the same primes. If you start 60 as 4 x 15 instead, the 4 gives 2 x 2 and the 15 gives 3 x 5, which is the same set of primes in a different order.

This is the method the calculator uses. Start with the smallest prime, 2, and divide it out as many times as it fits. Then move to the next candidate and repeat. For 100 the ladder looks like this. 100 / 2 = 50, 50 / 2 = 25, 25 / 5 = 5, 5 / 5 = 1. The divisors used were 2, 2, 5, and 5, so 100 = 2^2 x 5^2. Trial division only needs to test divisors up to the square root of the current remainder, because any factor larger than the square root would already have a smaller partner that was found earlier. When nothing divides the remainder, the remainder itself is prime.

Every whole number greater than 1 has exactly one prime factorization, apart from the order of the factors. This is the Fundamental Theorem of Arithmetic. It is the reason the factor tree and the ladder always agree with each other. It also makes the factorization a fingerprint of the number, because two numbers are equal exactly when their factorizations match.

Once you have the factorization, several useful counts follow without listing every divisor.

Counting Divisors

Each divisor is built by choosing an exponent for each prime, from 0 up to the exponent in the factorization. So the number of divisors is the product of (exponent + 1) over every prime. For 60 = 2^2 x 3 x 5, that is (2+1)(1+1)(1+1) = 12 divisors. They are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.

Euler's Totient

Euler's totient, written phi(n), counts how many integers from 1 to n share no common factor with n. From the factorization it equals the product of p^(a-1) x (p-1) for each prime. For 360 = 2^3 x 3^2 x 5, that gives 2^2 x (2-1) x 3 x (3-1) x (5-1) = 4 x 1 x 3 x 2 x 4 = 96. So 96 of the integers from 1 to 360 are coprime to 360.

Factorization is the master key for working with whole numbers. To simplify a fraction, divide the top and bottom by their greatest common divisor, which you can read straight from the factorizations. To add fractions with different denominators, you need the least common multiple, which comes from the same source. Our LCM and GCD calculator turns both of those jobs into one step. The same idea protects most encrypted web traffic. RSA cryptography multiplies two large primes to make a key, and its security rests on the fact that factoring the product back apart is extremely hard.

A quick lookup for the numbers people factor most often.