PrimeLab

Prime Factorization Calculator

Every number's unique breakdown into primes.

Number

Prime factorization

2^2 × 3 × 7

Factors (expanded): 2 × 2 × 3 × 7

What you entered

Number: 84

1
Divide repeatedly by the smallest possible prime
n = 84= 2 × 2 × 3 × 7
2
Group repeated primes into exponents
2 × 2 × 3 × 7= 2^2 × 3 × 7

Result

Prime factorization: 2^2 × 3 × 7

84 = 2^2 × 3 × 7 — this is its unique prime factorization (every whole number > 1 has exactly one).

Prime factorisation in expanded and exponent form

Every whole number above 1 breaks down into exactly one set of prime factors. The last column groups repeats as exponents.

Number	Prime factors	Exponent form
12	2, 2, 3	2^2 x 3
60	2, 2, 3, 5	2^2 x 3 x 5
84	2, 2, 3, 7	2^2 x 3 x 7
100	2, 2, 5, 5	2^2 x 5^2
360	2, 2, 2, 3, 3, 5	2^3 x 3^2 x 5
1024	2 repeated ten times	2^10

The fundamental theorem of arithmetic guarantees this decomposition is unique apart from the order, which is why prime factorisation is the backbone of number theory. 360 having factors 2, 3 and 5 in abundance is why it was adopted for degrees in a circle - it divides evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12 and more. 1024 as 2 to the tenth is why kilobytes were historically 1,024 bytes rather than 1,000. Factorisation is easy to verify and hard to reverse for very large numbers, which is precisely what RSA encryption relies on.

The fundamental theorem of arithmetic

Every integer greater than 1 has exactly one prime factorization — no matter how you break a number down, you always land on the same set of primes, just possibly in a different order.

Where exponent form comes from

When a prime repeats (like 84 = 2 × 2 × 3 × 7), grouping the repeats into a power — 2² × 3 × 7 — is the standard way factorizations are written in textbooks and used to quickly find a number's GCD, LCM, or total divisor count.

Frequently asked questions

How do I find the prime factorization of a number?

Divide by the smallest prime (2) repeatedly until it no longer divides. Move to the next prime (3, 5, 7...) and repeat. Stop when the quotient is 1. Example: 180 ÷ 2 = 90 ÷ 2 = 45 ÷ 3 = 15 ÷ 3 = 5 ÷ 5 = 1. So 180 = 2² × 3² × 5.

What is the fundamental theorem of arithmetic?

Every integer greater than 1 has exactly one prime factorization (up to the order of factors). 12 is always 2² × 3, never anything else. This uniqueness is why prime factorization is so useful — it's the number's fingerprint.

How does prime factorization help find GCD and LCM?

GCD: take each prime at its LOWEST power across both numbers. LCM: take each prime at its HIGHEST power. For 12 = 2²×3 and 18 = 2×3²: GCD = 2¹×3¹ = 6, LCM = 2²×3² = 36.