Prime Factorization Calculator

Break an integer into prime powers and explore its division ladder, positive divisors and divisor sum.

Calculation inputs

Precision & interpretation

The integer 1 has no prime factors. It has one positive divisor, 1, and its divisor sum is 1. Negative inputs and zero are outside this positive-integer tool.

Calculation results

Prime factorization2^2 × 3 × 784 · exact prime-power product

Positive divisors12

Sum of divisors224

Distinct prime factors3

Step-by-step solution

Choose the formula
$Divide n by prime factors until the quotient is 1.$
Substitute your inputs
$n = 84$
Calculate in order
$84 ÷ 2 = 42; 42 ÷ 2 = 21; 21 ÷ 3 = 7; 7 ÷ 7 = 1$
Write the result
$84 = 2^2 × 3 × 7$

Prime division ladder

Divide by 284 ÷ 2 = 42Divide by 242 ÷ 2 = 21Divide by 321 ÷ 3 = 7Divide by 77 ÷ 7 = 1

Every extracted divisor is prime; the final quotient is 1.

Positive divisors include 1 and n. Repeated prime factors are collected into powers.

Exact value & calculation details

2^2 × 3 × 7

Quantity	Value
2	Exponent 2
3	Exponent 1
7	Exponent 1

Understand the calculation

Concept & meaning

A positive integer greater than one has a unique prime-power factorization, apart from the order of its factors. The division ladder shows how those factors are extracted.

Formula & special cases

If n = ∏ pᵢᵉⁱ, its number of positive divisors is ∏(eᵢ + 1). The divisor sum is the product of 1 + p + … + pᵉ for every distinct prime.

Precision & limitations

The integer 1 has no prime factors. It has one positive divisor, 1, and its divisor sum is 1. Negative inputs and zero are outside this positive-integer tool.

n = \prod p_{i}^{e_{i}}

Worked example from your inputs

84 · exact prime-power product = 2^2 × 3 × 7

Frequently asked questions

Does 1 have a prime factorization?

It has an empty product of prime factors. It is neither prime nor composite.

Why are repeated factors written as powers?

An exponent records how many copies of the same prime occur in the product.

Do divisor counts include the number itself?

Yes. Positive divisors include 1 and n.

Is the factorization exact?

Yes, for the supported whole-number range. The prime factors multiply back to the input.

Show all questions (5 more)Show fewer questions

What does the Prime Factorization calculate?

Break an integer into prime powers and explore its division ladder, positive divisors and divisor sum.

Which values should I enter?

Use the labeled fields above. Whole number from 1 to 1,000,000,000,000. Values outside the supported range clear the result and show an explanation.

What formula does this calculator use?

n = ∏ pᵢᵉⁱ; d(n) = ∏(eᵢ + 1)

Why might my result differ from another source?

Results can differ when another source uses different units, rounding rules, conventions or assumptions. Compare the inputs and definitions before comparing results.

Does the result update automatically?

Yes. Inputs and presets update the result, teaching diagram and expandable steps together. Invalid input clears the previous result.

Sources & References

$OpenStax official logo$ OpenStax · Prime factorizationPrime factors and factor-tree methodsRead the reference $NIST official logo$ NIST DLMF · Number theoryPrime powers and divisor functionsRead the reference

References explain the mathematics; they do not imply institutional endorsement.