Detalles
- Factor primo más pequeño: —
- Factores primos: —
Paso a paso
Explicación:
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.
Escribe un entero como producto de factores primos.
Solo valores enteros. No se admiten decimales.
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: $$1 = 1$$
Keep exploring Mathematics or jump back to Calculators to browse more tools.
Preguntas sobre la calculadora de factorización prima
Respuestas rápidas sobre factores primos, potencias y casos especiales.
La factorización prima consiste en escribir un entero $n$ (con $n > 1$) como producto de números primos, normalmente en forma de potencias, por ejemplo $2^a \cdot 3^b \cdot 5^c$.
Sí. Para todo entero $n > 1$, la factorización prima es única salvo el orden de los factores (Teorema fundamental de la aritmética).
Divide $|n|$ repetidamente entre primos pequeños, registrando una cadena de divisiones (por ejemplo $84 = 2\cdot 42$), y después agrupa los primos repetidos en potencias.
La calculadora factoriza $|n|$ y muestra el resultado como $-1$ multiplicado por la factorización, por ejemplo $-84 = -1\cdot 2^2\cdot 3\cdot 7$.
El número $1$ no tiene factores primos, así que su factorización es $1$.
Porque $0$ admite infinitas descomposiciones (para cualquier entero $k$, $0 = 0\cdot k$), así que no existe una factorización prima única.
Con la confianza de miles de usuarios cada mes: herramientas rápidas, precisas y respetuosas con la privacidad.