Prime Factorization Calculator

What Is Prime Factorization?

Prime factorization is the process of expressing a positive whole number as a product of prime numbers, the numbers greater than 1 whose only factors are 1 and themselves (2, 3, 5, 7, 11, …). For example, 60 = 2 × 2 × 3 × 5, or 60 = 2² × 3 × 5.

However you find it, you always end up with the same primes. You might split 60 as 6 × 10 or as 4 × 15, but both lead to two 2s, one 3 and one 5. This is the Fundamental Theorem of Arithmetic: every whole number greater than 1 has exactly one prime factorization, apart from the order of the factors. That is why 1 is not counted as a prime: if it were, you could add any number of 1s, and the factorization would no longer be unique.

How to Find the Prime Factorization of a Number

The division method works for any number:

1. Start with the smallest prime that divides the number, usually 2.

2. Divide by that prime, and keep dividing by it while the result stays a whole number.

3. Move on to the next prime (3, 5, 7, …) and repeat with the quotient.

4. Stop when the quotient is 1.

5. Multiply all the primes you divided by, and combine repeated ones with exponents.

Example with 84: 84 ÷ 2 = 42, 42 ÷ 2 = 21, 21 ÷ 3 = 7, 7 ÷ 7 = 1. The divisors were 2, 2, 3 and 7, so 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7.

You only need to test primes up to the square root of the number that is left. If none divides it, what is left is prime.

Exponent Form

Repeated prime factors are easier to read with exponents: the exponent says how many times the prime is multiplied. 72 = 2 × 2 × 2 × 3 × 3 becomes 72 = 2³ × 3², and 1,024 = 2 × 2 × … × 2 (ten 2s) becomes 2¹⁰. A prime with no exponent, such as the 5 in 360 = 2³ × 3² × 5, appears once, so its exponent is 1.

Factors, Prime Factors and Factor Pairs

These three ideas are easy to mix up. Take 36:

Factors are all the whole numbers that divide 36 exactly: 1, 2, 3, 4, 6, 9, 12, 18 and 36.

Factor pairs are two factors whose product is 36: 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6.

Prime factors are only the factors that are prime: 2 and 3. The prime factorization multiplies them, with repeats, to make the number: 36 = 2² × 3².

Number of Factors From the Prime Factorization

If n = p^a × q^b × r^c …, then n has (a + 1)(b + 1)(c + 1)… positive factors. Each factor is built by choosing how many times to use each prime: p from 0 to a times (a + 1 choices), q from 0 to b times, and so on.

For 360 = 2³ × 3² × 5¹: (3 + 1)(2 + 1)(1 + 1) = 4 × 3 × 2 = 24 factors. For 36 = 2² × 3²: 3 × 3 = 9 factors. A number has an odd number of factors exactly when every exponent is even, which happens only for perfect squares.

Prime Factorization and Divisibility Rules

Divisibility rules tell you quickly which small primes to try first:

2: the number is even (its last digit is 0, 2, 4, 6 or 8).

3: the sum of its digits is divisible by 3. For 360, 3 + 6 + 0 = 9, so 3 divides 360.

5: the last digit is 0 or 5.

10: the last digit is 0, which means both 2 and 5 are prime factors.

The prime factorization answers every divisibility question at once: a number d divides n exactly when each prime in d appears in n at least as many times. 360 = 2³ × 3² × 5 is divisible by 12 = 2² × 3 but not by 16 = 2⁴.

Uses of Prime Factorization

Greatest common factor (GCF): multiply the primes the numbers share, each to its smallest power. 12 = 2² × 3 and 18 = 2 × 3², so GCF = 2 × 3 = 6.

Least common multiple (LCM): multiply every prime that appears, each to its largest power. LCM(12, 18) = 2² × 3² = 36.

Simplifying fractions: cancel the common primes. 36/60 = (2² × 3²) / (2² × 3 × 5) = 3/5.

Square roots and number theory: √360 = √(2² × 3² × 2 × 5) = 6√10, and the counting of factors, perfect squares and common denominators all start from the prime factorization. Large-number cryptography such as RSA relies on the fact that factoring a product of two huge primes is extremely slow.