Prime Factorization of 72

Every integer has exactly one prime fingerprint

The Fundamental Theorem of Arithmetic guarantees that every natural number greater than 1 can be written as a product of primes in exactly one way (ignoring order). This calculator finds that unique decomposition: type 360 and get 2^3 x 3^2 x 5. It also lists every divisor and counts them, which saves time when working with GCD, LCM, or fraction simplification.

Quick start

Type 84 and click "Factorize". Result: 84 = 2^2 x 3 x 7. Prime factors: 2, 2, 3, 7. Divisors: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 (12 total).

How the algorithm works

Trial division is the simplest factorization method. Start with the smallest prime (2) and keep dividing as long as the number is divisible. Then move to 3, then 5, 7, 11, and so on. When the quotient reaches 1, you have all the factors.

For 360: divide by 2 three times (360 -> 180 -> 90 -> 45), then by 3 twice (45 -> 15 -> 5), then by 5 once (5 -> 1). Result: 2^3 x 3^2 x 5.

Divisor count formula: If n = p1^a1 x p2^a2 x ... x pk^ak, then the number of divisors is (a1 + 1)(a2 + 1)...(ak + 1). For 360 = 2^3 x 3^2 x 5^1: divisors = (3+1)(2+1)(1+1) = 4 x 3 x 2 = 24.

Factorizations of common numbers

Number	Factorization	Divisors	Note
12	2^2 x 3	6	A dozen - remarkably many divisors for its size
60	2^2 x 3 x 5	12	Why clocks use 60 seconds/minutes
100	2^2 x 5^2	9	Only two distinct primes
360	2^3 x 3^2 x 5	24	Why a full circle has 360 degrees
1024	2^10	11	1 kilobyte in computing
7919	7919	2	The 1000th prime number - only 1 and itself
720720	2^4 x 3^2 x 5 x 7 x 11 x 13	240	Highly composite - 240 divisors from 6 primes

Practical examples

Simplifying fractions: To reduce 84/360, factorize both: 84 = 2^2 x 3 x 7 and 360 = 2^3 x 3^2 x 5. The GCD is 2^2 x 3 = 12. So 84/360 = 7/30.
Factorization makes GCD visible at a glance

Finding the LCM: LCM(12, 18) - factorize: 12 = 2^2 x 3 and 18 = 2 x 3^2. Take the highest power of each prime: 2^2 x 3^2 = 36. So LCM = 36.
LCM = product of highest prime powers from both numbers

Perfect squares: A number is a perfect square if every exponent in its factorization is even. 144 = 2^4 x 3^2 - both exponents are even, so 144 = 12^2. But 72 = 2^3 x 3^2 - the exponent of 2 is odd, so 72 is not a perfect square.
Check exponents: all even = perfect square

RSA cryptography: RSA security relies on the difficulty of factorizing large numbers. The product of two 300-digit primes is easy to compute, but reversing that multiplication would take billions of years with current hardware.
Factorization difficulty is the backbone of internet encryption

Gear ratios: A gear ratio of 84:60 simplifies to 7:5 (GCD = 12). Engineers factorize tooth counts to find the simplest ratio and check for common factors that could cause vibration patterns.
Mechanical engineering uses prime factors daily

Special number types revealed by factorization

Type	Definition	Example
Prime	Only 1 and itself as divisors	7, 13, 97, 7919
Perfect square	All exponents even	36 = 2^2 x 3^2, 144 = 2^4 x 3^2
Perfect cube	All exponents divisible by 3	216 = 2^3 x 3^3 = 6^3
Square-free	All exponents are 1	30 = 2 x 3 x 5, 42 = 2 x 3 x 7
Highly composite	More divisors than any smaller number	12, 24, 36, 48, 60, 120, 180, 360
Perfect number	Sum of proper divisors equals the number	6 = 1+2+3, 28 = 1+2+4+7+14

FAQ

What is the Fundamental Theorem of Arithmetic?

Every integer greater than 1 is either prime or can be written as a product of primes in exactly one way (up to the order of factors). This means 360 = 2^3 x 3^2 x 5 is the ONLY way to express 360 as a product of primes. There is no alternative factorization. This uniqueness is what makes prime factorization so useful.

Is 1 a prime number?

No. By modern convention, 1 is neither prime nor composite. It is the multiplicative identity. Excluding 1 from the primes is essential for the uniqueness of factorization - if 1 were prime, then 6 could be written as 2 x 3, 1 x 2 x 3, 1 x 1 x 2 x 3, and so on, destroying uniqueness.

How do you count divisors from the factorization?

Use the divisor count formula: if n = p1^a1 x p2^a2 x ... x pk^ak, then the number of divisors is (a1+1)(a2+1)...(ak+1). For 360 = 2^3 x 3^2 x 5^1: (3+1)(2+1)(1+1) = 4 x 3 x 2 = 24 divisors. Each divisor is a combination of prime factors with exponents from 0 to ai.

What is the largest prime number known?

As of early 2025, the largest known prime is 2^136,279,841 - 1, a Mersenne prime with over 41 million digits. It was discovered by the Great Internet Mersenne Prime Search (GIMPS). There are infinitely many primes (proved by Euclid around 300 BC), so the record will keep being broken.

How is prime factorization used in cryptography?

RSA encryption relies on the fact that multiplying two large primes (say, 300 digits each) is fast, but reversing the operation - finding which two primes were multiplied - is computationally infeasible with current technology. A 600-digit product would take billions of years to factorize. This asymmetry secures online banking, HTTPS, and digital signatures.

How do I use factorization to find the GCD?

Factorize both numbers, then multiply the shared prime factors with the lower exponent. GCD(84, 360): 84 = 2^2 x 3 x 7, 360 = 2^3 x 3^2 x 5. Shared primes: 2 (min exponent 2) and 3 (min exponent 1). GCD = 2^2 x 3 = 12. For the LCM, use the higher exponent of every prime: 2^3 x 3^2 x 5 x 7 = 2520.

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