Updated Number Theory Suite

Prime Factorization Calculator

Find prime factors, all factors, factorization steps, GCD, LCM, Euler’s totient, sum of divisors, and more. Includes prime checker and number theory tools in one place.

Prime Factors All Factors GCD & LCM Totient & Divisor Sums

All-in-One Prime Factorization & Number Theory Calculator

Switch between prime factorization, factors, step-by-step breakdowns, advanced divisor functions, GCD/LCM, and prime checking.

Recommended range: 2 to 1,000,000,000 (1e9). Larger values may take longer.
Optional. Leave 0 to auto-check n±2.

Prime Factorization Calculator – Break Any Number Into Primes

This Prime Factorization Calculator breaks down any positive integer into its prime components and shows every related number-theory result: all factors, GCD, LCM, Euler’s totient, divisor functions, and full factorization steps. Whether you're analyzing integers for math class, programming, cryptography, or number-theory exploration, this tool gives you everything in one place.

What Is Prime Factorization?

Prime factorization is the process of expressing a number as a product of prime numbers. Every integer greater than 1 has a unique factorization according to the Fundamental Theorem of Arithmetic.

360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 = 2^{3} \times 3^{2} \times 5

This calculator uses optimized trial division to produce the factor list and exponent form, even for large values up to 1 billion.

How Prime Factorization Works

The factorization method is straightforward but powerful:

  • Divide by the smallest prime \(2\) until it no longer divides.
  • Check primes \(3, 5, 7, 11, \ldots\)
  • Stop once the divisor exceeds \(\sqrt{n}\).

If the remaining number is greater than 1, it is itself a prime.

Prime Factorization Formulas

  • Prime factorization of n:
    n = p_1^{e_1} \; p_2^{e_2} \; p_3^{e_3} \dots
  • Total number of factors:
    d(n) = (e_1 + 1)(e_2 + 1)(e_3 + 1)\dots
  • Sum of divisors:
    \sigma(n) = \prod_{i=1}^{k} \frac{p_i^{\,e_i+1}-1}{p_i - 1}
  • Euler's Totient Function:
    \varphi(n) = n \prod_{i=1}^{k} \left(1 - \frac{1}{p_i}\right)

Finding All Factors

Once the prime factorization is known, all factors can be generated by considering every combination of prime powers:

\text{If } n = p_1^{e_1} p_2^{e_2},\text{ then factors} = p_1^{0 \ldots e_1} \times p_2^{0 \ldots e_2}

The calculator automatically lists all factors in ascending order.

GCD and LCM Using Prime Factors

Greatest Common Divisor (GCD)

\text{GCD}(a, b) = \prod p_i^{\,\min(e_{i,a}, e_{i,b})}

Least Common Multiple (LCM)

\text{LCM}(a, b) = \prod p_i^{\,\max(e_{i,a}, e_{i,b})}

The tool also confirms the identity:

\text{GCD}(a,b)\cdot\text{LCM}(a,b) = a \cdot b

Prime Checker and Twin Primes

The Prime Tools tab lets you:

  • Check if a number is prime
  • Find twin primes such as (11, 13) or (17, 19)
  • Compute fast modular arithmetic (a⁽ᵇ⁾ mod m)

Advanced Number Classification

The calculator uses the divisor sum \(\sigma(n)\) to classify integers:

  • Perfect: \(\sigma(n) = 2n\)
  • Abundant: \(\sigma(n) > 2n\)
  • Deficient: \(\sigma(n) < 2n\)

Examples:

  • Perfect: 6, 28
  • Abundant: 12, 18, 20
  • Deficient: 8, 14, 15

Why Prime Factorization Matters

Prime factorization appears everywhere:

  • Fractions: simplifying ratios
  • Algebra: recognizing patterns
  • Cryptography: RSA depends on prime difficulty
  • Computer science: hash functions, modular arithmetic
  • Number theory research: primes define structure

Example Prime Factorizations

Number Prime Factorization
120 \(2^{3} \times 3 \times 5\)
450 \(2 \times 3^{2} \times 5^{2}\)
999 \(3^{3} \times 37\)

How to Use This Prime Factorization Calculator

  • Enter any positive integer
  • Get prime factors in list and exponent form
  • Switch to other tabs to see steps, all factors, totient, GCD/LCM, and more

Try More Math Tools

MyTimeCalculator offers dozens of number-theory, arithmetic, algebra, geometry, and unit-conversion tools. Use this Prime Factorization Calculator along with our GCF, LCM, prime checker, percentage, and statistics calculators to speed up homework, research, and daily calculations.

Prime Factorization Calculator – FAQ

What is prime factorization?

Prime factorization expresses a number as a product of prime numbers. For example, \( 48 = 2^4 \times 3 \).

How do I find prime factors?

You repeatedly divide by the smallest primes (2, 3, 5, 7…) until the number becomes 1. The calculator does this automatically.

What is the difference between factors and prime factors?

Factors include all divisors of a number. Prime factors only include the prime-number components.

How can I compute the number of divisors?

The formula is \( d(n) = (e_1+1)(e_2+1)\dots \), where \( e_i \) are the exponents in the prime factorization.

What is Euler’s Totient function?

It counts the positive integers less than \( n \) that are coprime to \( n \):

\varphi(n)=n\prod (1-1/p).

What is a perfect number?

A perfect number satisfies \( \sigma(n) = 2n \). Examples include 6 and 28.

Does the calculator check for twin primes?

Yes. It automatically tests \( n-2 \) and \( n+2 \) to see if they form a twin prime pair.

Can I factor very large numbers?

The tool supports up to 1 billion (1e9). Beyond that, prime factorization becomes computationally expensive.

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