Details
- Smallest prime factor: —
- Prime factors: —
Step-by-step
Explanation:
Prime Factorization Calculator
This tool finds the prime factorization of an integer and shows the work step by step using a division chain, then groups repeated factors into powers. Enter $n$ (negatives are allowed) to factorize $|n|$ and, when needed, show the result as $-1$ times the factorization. For more number-structure topics, explore Number Theory.
Write an integer as a product of prime factors.
Integer values only. Decimals are not allowed.
What Is Prime Factorization?
Prime factorization writes an integer as a product of prime numbers. For any integer $n \gt 1$, the factorization is unique (up to the order of the factors). Results are usually shown in power form, like $2^3 \cdot 3^2 \cdot 5$, which is compact and easy to read.
Why Prime Factorization Matters
Prime factors help you understand a number’s structure and make many number theory tasks easier. For example, primes tell you whether a number is factorable at all, and shared prime factors explain why numbers have a nontrivial GCD.
- Prime vs composite: confirm whether a number has nontrivial factors using the Prime Checker.
- Common factors: relate factorizations to the GCD Calculator when comparing two integers.
How Prime Factorization Is Calculated
The calculator factors $|n|$ by repeatedly dividing by small primes. Each successful division is recorded as a division chain (for example, $84 = 2 \cdot 42$). After the chain is built, repeated primes are collected into powers to produce the final compact form.
$$n=\prod p_i^{e_i}$$
Input Rules and Edge Cases
- Negative input: the tool factors $|n|$ and reports the result as $-1$ times the factorization.
- $n=1$: has no prime factors, so the factorization is $1$.
- $n=0$: prime factorization is not defined.
- Very large integers: results can exceed safe integer range in the browser.
Worked Examples With Steps
Example 1: Composite number with powers
Factor a composite number and collect repeated primes into powers.
Find: $$360$$
Division chain: $$360 = 2\cdot 180$$ $$180 = 2\cdot 90$$ $$90 = 2\cdot 45$$ $$45 = 3\cdot 15$$ $$15 = 3\cdot 5$$
Collect factors: $$360 = 2\cdot 2\cdot 2\cdot 3\cdot 3\cdot 5$$ $$360 = 2^3\cdot 3^2\cdot 5$$
So: $$360 = 2^3\cdot 3^2\cdot 5$$
Example 2: Prime input
If $n$ is prime, its prime factorization is itself.
Find: $$97$$ Since $97$ is prime: $$97 = 97$$
Example 3: Negative input
The tool factors $|n|$ and includes a factor of $-1$ in the final result.
Find: $$-84$$ Work with: $$|{-84}|=84$$
Division chain: $$84 = 2\cdot 42$$ $$42 = 2\cdot 21$$ $$21 = 3\cdot 7$$
Collect factors and apply the sign: $$84 = 2^2\cdot 3\cdot 7$$ $$-84 = -1\cdot 2^2\cdot 3\cdot 7$$
Example 4: $n=1$
The number $1$ has no prime factors.
Find: $$1$$ So: $$1 = 1$$
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Questions About the Prime Factorization Calculator
Quick answers about prime factors, powers, and edge cases.
Prime factorization is writing an integer $n$ (with $n > 1$) as a product of prime numbers, typically in power form like $2^a \cdot 3^b \cdot 5^c$.
Yes. For every integer $n > 1$, the prime factorization is unique up to the order of the factors (the Fundamental Theorem of Arithmetic).
It repeatedly divides $|n|$ by small primes, recording a division chain (for example $84 = 2\cdot 42$), then groups repeated primes into powers.
The calculator factorizes $|n|$ and reports the result as $-1$ times the prime factorization, for example $-84 = -1\cdot 2^2\cdot 3\cdot 7$.
The number $1$ has no prime factors, so its factorization is $1$.
Because $0$ has infinitely many factorizations (for any integer $k$, $0 = 0\cdot k$), so there is no unique prime factorization.
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