Mathematics CalcyZone Calculator

Prime Factorization Calculator

Q: What is prime factorization?

Prime factorization breaks down a composite number into a product of prime numbers.

Q: Is 1 a prime number?

No. Primes have exactly two divisors, and 1 has only one.

Q: Why does factorisation matter in computing?

Multiplying large primes is easy but factoring their product is very hard, which underpins RSA encryption.

Decompose any integer into its unique prime factor product tree.

Enter Integer

Calculation Solution

Prime Factors: 2 × 2 × 5 × 5

Product: 100

Mathematical Formula

N = p1^a * p2^b * ... * pk^m

Overview & Explanation

The Fundamental Theorem of Arithmetic states that every integer greater than 1 is either prime or can be uniquely represented as a product of prime numbers.

How It Works

Divide the number by the smallest prime factor (2).
Continue dividing by primes (3, 5, 7...) until the quotient is 1.

Practical Applications

Simplifying radicals and roots.
Solving encryption algorithms.

Breaking a number into primes

Every whole number greater than 1 is either prime or can be written as a product of primes in exactly one way, ignoring order. This is the fundamental theorem of arithmetic. For 360, divide repeatedly by the smallest prime that fits: 360 ÷ 2 = 180, ÷ 2 = 90, ÷ 2 = 45, ÷ 3 = 15, ÷ 3 = 5, and 5 is prime. So 360 = 2³ × 3² × 5.

What the factorisation tells you

The number of divisors comes straight from the exponents: add 1 to each and multiply. For 2³ × 3² × 5¹ that is (3 + 1)(2 + 1)(1 + 1) = 24 divisors, which is one reason 360 was chosen for degrees in a circle.

A number is a perfect square when every exponent is even (144 = 2⁴ × 3²) and a perfect cube when every exponent is a multiple of 3. Factorisations also give GCDs and LCMs instantly and let you simplify square roots: √360 = √(2² × 3² × 10) = 6√10.

Testing for primes

You only need to test divisors up to the square root of a number. To check whether 97 is prime, try 2, 3, 5 and 7 (since √97 < 10); none divide it, so 97 is prime. Quick divisibility rules help: a number is divisible by 3 if its digits add to a multiple of 3, and by 5 if it ends in 0 or 5.