What is prime factorization?

It writes a whole number above 1 as a product of primes. A prime has exactly two positive divisors: 1 and itself. For example, 1,386 = 2 × 3^2 × 7 × 11. The prime factors are unique apart from their order.

How do I use this prime factorization calculator?

Type the number in the Number field. The result updates as you type. Choose Exponents, Product or Tree to change the leading view, then choose Copy for a plain-text answer. For example, entering 729 gives 3^6, while 2024 gives 2^3 × 11 × 23 and 9604 gives 2^2 × 7^4.

How do I read the factor tree?

Each split places the smallest prime factor on the left and the remaining quotient on the right. For 1,386, the quotients are 693, 231, 77 and 11. Collect the prime leaves to get 2 × 3 × 3 × 7 × 11. A prime input has a single-node tree.

How do I use the prime factorization ladder method?

The Division ladder below the results divides the current number by one prime at each step until the quotient is 1. For 1386, the rows are 1386 ÷ 2 = 693, 693 ÷ 3 = 231, 231 ÷ 3 = 77, 77 ÷ 7 = 11 and 11 ÷ 11 = 1. Repeated primes each get a row. For a prime such as 97, the only row is 97 ÷ 97 = 1.

How many positive divisors does a number have?

Add 1 to each prime exponent, then multiply. For 1,386 = 2^1 × 3^2 × 7^1 × 11^1, the count is (1 + 1) × (2 + 1) × (1 + 1) × (1 + 1) = 24. The Divisors readout includes 1 and the number itself.

How can I find a square root by prime factorization without a calculator?

Write the prime factors and group equal factors into pairs. Take one factor from each pair outside the square root and leave unpaired factors inside. For example, 72 = 2 × 2 × 2 × 3 × 3. The pairs of 2 and 3 give sqrt(72) = 2 × 3 × sqrt(2) = 6sqrt(2); the unpaired 2 stays inside. If every factor is paired, the square root is a whole number. This is a manual method; the calculator shows integer factors rather than a simplified square root.

What numbers are accepted?

Enter a whole number from 2 through 10,000,000,000,000,000. Correctly grouped commas such as 1,386 are accepted; 1,5 is rejected. Zero, 1, negative numbers, decimals, fractions and variables are refused with a specific message. The calculator does not round invalid inputs.

Can it factor large numbers exactly?

Within the stated input limit, it uses BigInt integer arithmetic and exact trial division. For example, 9,007,199,254,740,993 factors as 3 × 107 × 28,059,810,762,433. Large primes can take longer than numbers with small factors, and speed depends on the device.

Why does 1 have no prime factors?

1 is not prime. Including it as a prime factor would let any product include arbitrarily many extra factors of 1, so prime factorization would lose its uniqueness. Zero has no finite prime factorization because every prime divides it.

Is this a browser program, and is my number saved?

Yes. This free calculator runs in a browser on a phone, tablet or computer without an installation or account. The number and chosen view are remembered locally after you change them. Start over clears the saved draft. Copy writes the factors and divisor count to your clipboard when browser access is available.

Prime Factorization Calculator

Type a whole number above 1 to read its prime factorization as exponent form, a plain product and a factor tree, with a division ladder showing each step. The calculator counts positive divisors and copies the result as text. Calculations use exact integer arithmetic for numbers up to 10,000,000,000,000,000.

1,386 factors into2 × 3² × 7 × 112 × 3 × 3 × 7 × 11

Number

Show

Divisors

1,386 = 2 × 3^2 × 7 × 11, written out 2 × 3 × 3 × 7 × 11. It has 24 divisors, from (1 + 1) × (2 + 1) × (1 + 1) × (1 + 1) = 24.

Whole numbers above 1, up to ten thousand million million. Thousands may be grouped in threes, as 1,386.

Division ladder

Divide by one prime at each step, repeating it as needed, until the quotient is 1.

1386 ÷ 2 = 693
693 ÷ 3 = 231
231 ÷ 3 = 77
77 ÷ 7 = 11
11 ÷ 11 = 1

A prime97
97 is prime
A power of two1024
2^10 · 11 divisors
A 19-digit numberpast quick trial division
That number is above ten thousand million million (10,000,000,000,000,000), which is where trial division stops being quick. Type a whole number of 10,000,000,000,000,000 or less and it will factorize in a moment.

What the tree and the exponents each tell you. Every whole number above 1 is a product of primes in exactly one way, which is why there is a single right answer here. The tree is the working: take out the smallest prime that divides the number, put it on the left, carry what is left down the right, and stop when the right-hand side is itself prime. Collecting the repeats gives the exponent form, and the exponent form gives the divisor count for free: add one to each exponent and multiply, because a divisor picks each prime anywhere from none of it up to all of it. For 1,386 that is (1 + 1) × (2 + 1) × (1 + 1) × (1 + 1) = 24. Exact for every whole number it accepts: the factors are found by trial division in exact integer arithmetic and multiply back to the number you typed. Above ten thousand million million it says the number is too large rather than guessing, because trial division stops being quick there.

Common questions

What is prime factorization?: It writes a whole number above 1 as a product of primes. A prime has exactly two positive divisors: 1 and itself. For example, 1,386 = 2 × 3^2 × 7 × 11. The prime factors are unique apart from their order.
How do I use this prime factorization calculator?: Type the number in the Number field. The result updates as you type. Choose Exponents, Product or Tree to change the leading view, then choose Copy for a plain-text answer. For example, entering 729 gives 3^6, while 2024 gives 2^3 × 11 × 23 and 9604 gives 2^2 × 7^4.
How do I read the factor tree?: Each split places the smallest prime factor on the left and the remaining quotient on the right. For 1,386, the quotients are 693, 231, 77 and 11. Collect the prime leaves to get 2 × 3 × 3 × 7 × 11. A prime input has a single-node tree.
How do I use the prime factorization ladder method?: The Division ladder below the results divides the current number by one prime at each step until the quotient is 1. For 1386, the rows are 1386 ÷ 2 = 693, 693 ÷ 3 = 231, 231 ÷ 3 = 77, 77 ÷ 7 = 11 and 11 ÷ 11 = 1. Repeated primes each get a row. For a prime such as 97, the only row is 97 ÷ 97 = 1.
How many positive divisors does a number have?: Add 1 to each prime exponent, then multiply. For 1,386 = 2^1 × 3^2 × 7^1 × 11^1, the count is (1 + 1) × (2 + 1) × (1 + 1) × (1 + 1) = 24. The Divisors readout includes 1 and the number itself.
How can I find a square root by prime factorization without a calculator?: Write the prime factors and group equal factors into pairs. Take one factor from each pair outside the square root and leave unpaired factors inside. For example, 72 = 2 × 2 × 2 × 3 × 3. The pairs of 2 and 3 give sqrt(72) = 2 × 3 × sqrt(2) = 6sqrt(2); the unpaired 2 stays inside. If every factor is paired, the square root is a whole number. This is a manual method; the calculator shows integer factors rather than a simplified square root.
What numbers are accepted?: Enter a whole number from 2 through 10,000,000,000,000,000. Correctly grouped commas such as 1,386 are accepted; 1,5 is rejected. Zero, 1, negative numbers, decimals, fractions and variables are refused with a specific message. The calculator does not round invalid inputs.
Can it factor large numbers exactly?: Within the stated input limit, it uses BigInt integer arithmetic and exact trial division. For example, 9,007,199,254,740,993 factors as 3 × 107 × 28,059,810,762,433. Large primes can take longer than numbers with small factors, and speed depends on the device.
Why does 1 have no prime factors?: 1 is not prime. Including it as a prime factor would let any product include arbitrarily many extra factors of 1, so prime factorization would lose its uniqueness. Zero has no finite prime factorization because every prime divides it.
Is this a browser program, and is my number saved?: Yes. This free calculator runs in a browser on a phone, tablet or computer without an installation or account. The number and chosen view are remembered locally after you change them. Start over clears the saved draft. Copy writes the factors and divisor count to your clipboard when browser access is available.

Exact for every whole number it accepts: the factors are found by trial division in exact integer arithmetic and multiply back to the number you typed. Above ten thousand million million it says the number is too large rather than guessing, because trial division stops being quick there.