Prime Factorization Calculator
Break down numbers into prime factors, check if numbers are prime, find all divisors, and list primes within a range. Supports numbers up to 1 trillion.
Prime Factorization
Break down a number into its prime factors
How to Use the Prime Factorization Calculator
Select the type of calculation you need, enter your number(s), and click Calculate. The calculator provides four different modes for various prime number operations.
Calculator Modes
Prime Factorization
Breaks down any positive integer into its prime factors. For example, 360 = 2^3 × 3^2 × 5. The result shows each prime factor with its exponent (how many times it divides the number).
Prime Number Check
Determines whether a number is prime. A prime number is only divisible by 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23...
All Divisors
Finds every number that divides evenly into the given number. Also calculates the count and sum of all divisors. For example, divisors of 12 are: 1, 2, 3, 4, 6, 12.
Primes in Range
Lists all prime numbers between two values. Useful for finding primes in a specific range, like all primes between 1 and 100.
What Are Prime Numbers?
A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. The first few primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47...
Why Prime Factorization Matters
- Finding GCD (Greatest Common Divisor) and LCM (Least Common Multiple)
- Simplifying fractions to lowest terms
- Cryptography and computer security (RSA encryption)
- Understanding number properties and relationships
- Solving divisibility problems
Fundamental Theorem of Arithmetic
Every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. This is why prime factorization is called the "DNA" of a number.
Worked Example: Factoring 360
Divide by the smallest prime repeatedly: 360 ÷ 2 = 180, 180 ÷ 2 = 90, 90 ÷ 2 = 45. Now 45 is odd, so try 3: 45 ÷ 3 = 15, 15 ÷ 3 = 5, and 5 is itself prime. Collecting the divisors gives 360 = 2³ × 3² × 5. From the exponents you can also count the divisors: (3+1)(2+1)(1+1) = 24 divisors.
Frequently Asked Questions
How do I find the prime factorization of a number?+
Enter any positive integer and choose the Prime Factorization mode. The calculator repeatedly divides by primes and returns the result in exponent form, such as 360 = 2^3 × 3^2 × 5. It handles numbers up to 1 trillion, which is far beyond what is practical to factor by hand.
How can I check if a large number is prime?+
Use the Prime Check mode: enter the number and the calculator tests it against all candidate divisors up to its square root. If nothing divides it evenly, the number is prime. For example, 997 is prime, while 1,001 = 7 × 11 × 13 is not — a fact easy to miss by inspection.
What is the largest number this calculator can factor?+
The calculator supports integers up to 1 trillion (10^12). Trial division only needs to test divisors up to the square root of the number — at most 1 million candidates — so results come back almost instantly even at the upper limit.
Why is 1 not a prime number?+
A prime must have exactly two distinct divisors: 1 and itself, and the number 1 has only one divisor. Excluding 1 also preserves the Fundamental Theorem of Arithmetic — if 1 were prime, factorizations would no longer be unique, since you could insert extra factors of 1 endlessly.
Is this prime factorization calculator free and does it work offline?+
Yes, it is completely free with no sign-up. All factoring runs entirely in your browser using JavaScript, so no data is sent to a server, and once the page is loaded the calculations work even with a spotty connection. It also works on mobile phones and tablets.
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