Prime Factorization of 144
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Prime Factorization
2^4 × 3^2
Summary
144 = 2^4 × 3^2, giving it 15 total divisors (sum of all divisors: 403).
Expanded Form
2 × 2 × 2 × 2 × 3 × 3
Total Divisors
15
Prime factors and how many times each is used
What is Prime Factorization?
Prime factorization breaks a whole number down into the unique set of prime numbers that multiply together to produce it. Every whole number greater than 1 has exactly one prime factorization (up to the order of the factors) — a fact known as the Fundamental Theorem of Arithmetic — which makes prime factorization a foundational tool throughout number theory.
A prime number is any whole number greater than 1 whose only positive divisors are 1 and itself (2, 3, 5, 7, 11, …). Any number that isn't prime is called composite, and every composite number can be uniquely broken down into primes, often written in exponent form — for example, 360 = 2³ × 3² × 5.
How Prime Factorization Works
The standard method (trial division) repeatedly divides the number by the smallest possible prime until it no longer divides evenly, then moves to the next prime, continuing until only 1 remains. For 360: divide by 2 three times to get 45, divide by 3 twice to get 5, then divide by 5 once to get 1 — giving 360 = 2 × 2 × 2 × 3 × 3 × 5, or 2³ × 3² × 5 in exponent form.
Counting and Summing Divisors from the Factorization
Once you have a number's prime factorization, you can compute its total number of divisors without listing them all: add 1 to each prime's exponent and multiply the results together. For 360 = 2³ × 3² × 5¹, that's (3+1) × (2+1) × (1+1) = 4 × 3 × 2 = 24 total divisors. A similar formula (summing each prime's geometric series of powers) gives the sum of all divisors without listing them.
Prime Factorization vs. the Factor Calculator
Prime factorization decomposes a number into primes only — the smallest possible building blocks. The Factor Calculator instead lists every divisor of a number, prime or not (for 360, that includes 1, 2, 3, 4, 5, 6, …, 360 — all 24 of them). The two are related: every divisor of a number can be built by combining a subset of its prime factors.
Why Prime Factorization Matters
Prime factorization underlies simplifying fractions, finding the greatest common factor (GCF) and least common multiple (LCM) of two numbers, and reducing radicals in algebra. It's also the mathematical basis of RSA public-key cryptography, which relies on the fact that multiplying two large primes together is fast, while factoring the resulting product back into those primes is computationally infeasible for sufficiently large numbers.
Example — Your Current Input
144 = 2^4 × 3^2, giving it 15 total divisors (sum of all divisors: 403).
Additional Example — Factoring 84
84 is even, so divide by 2 to get 42, then divide by 2 again to get 21 (2² so far). 21 isn't even, so move to 3: 21 ÷ 3 = 7. Finally, 7 is itself prime, so the process stops. That gives 84 = 2² × 3 × 7, with (2+1) × (1+1) × (1+1) = 12 total divisors: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.
About This Parameter
- Number
- Any whole number of 2 or greater. Prime numbers will show their own factorization (just themselves), while composite numbers break down into two or more prime factors, possibly repeated.
Frequently Asked Questions
Why isn't 1 considered prime?
By definition, a prime number has exactly two distinct positive divisors: 1 and itself. The number 1 has only one divisor (itself), so it fails that definition. Excluding 1 also preserves the Fundamental Theorem of Arithmetic's uniqueness guarantee — if 1 counted as prime, every number would have infinitely many "factorizations" by tacking on extra 1s.
Is every prime factorization unique?
Yes — this is precisely what the Fundamental Theorem of Arithmetic guarantees. Every whole number greater than 1 has exactly one prime factorization, aside from the order in which the factors are written.
How do I know if a large number is prime?
Trial division (checking every integer up to the square root of the number) works but gets slow for very large numbers. This calculator uses trial division and works quickly for numbers up to about 10^15; for cryptographically large numbers (hundreds of digits), specialized primality tests are required instead.
Why does trial division only need to check up to the square root?
If a number n has a factor larger than its square root, it must also have a corresponding factor smaller than the square root (since their product equals n). So once you've checked every candidate up to √n without finding a factor, there's no larger factor left to find — the number must be prime.