Prime Factorization Calculator

1. Enter a whole number greater than 1.
2. Read the prime factorization in exponent form.
3. See the number of divisors the number has.
4. Check whether the number is prime.

Here is what each result means:

Prime factorization breaks a whole number into a product of prime numbers. A prime is a number greater than 1 whose only divisors are 1 and itself, such as 2, 3, 5 and 7. Every whole number above 1 has exactly one prime factorization, a rule known as the fundamental theorem of arithmetic.

For example, 360 = 2^3 x 3^2 x 5. Writing repeated primes as powers keeps the answer compact. Prime factorization underlies simplifying fractions, finding a greatest common divisor or least common multiple, and much of cryptography.

It uses trial division, dividing by the smallest prime that fits, again and again, until only 1 is left.

1. Divide out every factor of 2, then every 3, then odd numbers upward.
2. Stop when the divisor squared passes the remaining number.
3. Whatever is left, if above 1, is itself a prime factor.

Once you have the factors, the greatest common divisor calculator and LCM calculator use them directly.

Factor 360 into primes.

Calculation: 360 divided by 2 is 180, then 90, then 45, so three 2s. Then 45 divided by 3 is 15, then 5, so two 3s. Finally 5 is prime. That gives 360 = 2^3 x 3^2 x 5. The number of divisors is (3+1)(2+1)(1+1) = 24.

Trial division works through the primes in order. The first several are worth knowing.

2 is the only even prime; every other even number has 2 as a factor, so it is not prime.

Prime factors are the building blocks other number-theory tools rely on.

So once a number is factored, many other problems become quick to solve by hand.

The Exponents in the Factorization

The divisor count is the product of one more than each exponent, so higher powers mean many more divisors.

How Many Distinct Primes

More different prime factors multiply the divisor count further, even at low powers.

Being Prime

A prime has exactly two divisors, 1 and itself, the fewest possible above 1.

Simplifying Fractions

Factor the top and bottom to cancel shared primes cleanly.

Finding a GCD or LCM

Compare the prime factorizations of two numbers to combine them.

Homework and Number Theory

Check a factor tree, or test whether a number is prime.

1. Listing 1 as a Prime

1 is not a prime number. Prime factorizations use primes from 2 upward.

2. Stopping Too Early

Keep dividing until only 1 remains. A leftover above 1 is still a prime factor.

3. Forgetting Exponents

Repeated primes should be written as powers, so 8 is 2 cubed, not just 2.

4. Missing a Factor of 2

Divide out every 2 before moving on, or an even number will look wrong.

5. Confusing Factors with the Factorization

The divisors of a number are not the same as its prime factors; primes are the building blocks.

The method is exact for whole numbers in range; only very large inputs slow it down.

It is breaking a whole number into the prime numbers that multiply to make it. For example, 360 is 2 cubed times 3 squared times 5. Every whole number above 1 has exactly one such factorization.

Divide by the smallest prime that fits, again and again, moving up through 2, 3, 5 and so on until only 1 remains. Write repeated primes as powers. This tool does it automatically.

No. A prime is a number greater than 1 whose only divisors are 1 and itself. Since 1 has only one divisor, it is not counted as prime.

If trial division finds no prime factor up to its square root, the number is prime. This tool reports whether your number is prime as one of its results.

Add one to each exponent in the prime factorization and multiply the results. For 360, which is 2 cubed times 3 squared times 5, that is (3+1)(2+1)(1+1) = 24 divisors.

Powers keep the answer compact and make patterns clear. Writing 8 as 2 cubed is shorter than 2 times 2 times 2, and it feeds directly into divisor and GCD formulas.

It simplifies fractions, finds greatest common divisors and least common multiples, and underpins cryptography, where the difficulty of factoring very large numbers keeps data secure.

It handles numbers up to about a trillion quickly. Much larger numbers are declined here, because factoring big numbers is genuinely hard, which is exactly why it protects encryption.

No. The calculation runs in your browser and nothing you enter is stored or sent anywhere, unless you choose Save, which keeps the result only on this device.

• Integer factorization (Wikipedia).
• Prime factorization explained (Maths Is Fun).
• Fundamental theorem of arithmetic (Wikipedia).