Factors Calculator – Find All Divisors & Prime Factors of a Number

Factors Finder

Enter a positive integer

The Factors Calculator finds all positive divisors (factors) of any integer. It also shows factor pairs, prime factorisation, and the total number of factors. Whether you need to factor a number for algebra, simplify fractions, find greatest common divisors (GCD), or just explore divisibility, this tool provides instant results with a complete step‑by‑step explanation.

What Are Factors?

A factor (or divisor) of a whole number n is another whole number that divides n exactly with no remainder. For example, the factors of 15 are 1, 3, 5, and 15 because 15 ÷ 1 = 15, 15 ÷ 3 = 5, 15 ÷ 5 = 3, 15 ÷ 15 = 1. Every number has at least two factors: 1 and itself.

Methods to Find Factors

1. Trial division: Test each integer from 1 to √n. If i divides n, then both i and n/i are factors.

2. Prime factorisation first: Find the prime factors, then generate all combinations to list factors.

3. Using factor pairs: List pairs (i, n/i) for i up to √n.

Real‑World Applications of Factors

Simplifying fractions: Finding common factors between numerator and denominator.
Finding greatest common divisor (GCD): The largest factor common to two numbers.
Algebra and number theory: Factoring polynomials, solving Diophantine equations.
Everyday grouping: Dividing items into equal groups (e.g., 15 candies can be shared among 1, 3, 5, or 15 people).

Understanding Prime Factorisation

Every integer greater than 1 can be uniquely expressed as a product of prime numbers. This is called the Fundamental Theorem of Arithmetic. For example, 15 = 3 × 5. Prime factorisation is the key to many number theory problems, including finding the LCM, GCD, and the total number of factors: if n = p₁ᵃ × p₂ᵇ × … then the number of factors = (a+1)(b+1)… .

Factor Table for Common Numbers

Number	Factors
12	1,2,3,4,6,12
15	1,3,5,15
24	1,2,3,4,6,8,12,24
36	1,2,3,4,6,9,12,18,36
48	1,2,3,4,6,8,12,16,24,48
100	1,2,4,5,10,20,25,50,100

Step‑by‑Step Example: Factors of 15

To find factors of 15: test divisibility from 1 to √15 ≈ 3.87.
– 1 divides 15 → include 1 and 15/1 = 15 → factors 1,15.
– 2 does not divide 15 (15 ÷ 2 = 7.5).
– 3 divides 15 → include 3 and 15/3 = 5 → factors 3,5.
– Stop because 4 greater than √15.
Sorted: 1,3,5,15.
Factor pairs: 1×15, 3×5. Prime factorisation: 15 = 3 × 5.

Common Mistakes When Finding Factors

Missing factor pairs: For each divisor i ≤ √n, don’t forget to include n/i.
Including 0 or negative numbers: Factors are positive for most basic applications.
Not sorting: Listing unsorted factors can be confusing – our calculator sorts them automatically.
Confusing factors with multiples: Factors divide the number; multiples are the result of multiplying the number by an integer.

Use this factors calculator for homework, algebra, or any number exploration. The step‑by‑step output clearly shows how factors are derived, making it an excellent learning resource.

Step‑by‑Step Manual Example (15)

Input: 15
Test i=1 → 15÷1=15 → factors 1,15
i=2 → not divisible
i=3 → 15÷3=5 → factors 3,5
i=4 greater tahn √15 → stop.
Final factors: 1,3,5,15

Frequently Asked Questions about Factors

What are factors of a number?

Factors (or divisors) are integers that divide the given number exactly without leaving a remainder. For example, factors of 12 are 1,2,3,4,6,12.

How do you find factors of a number?

You can test divisibility from 1 up to the square root of the number. For each divisor i, include i and n/i. Our calculator does this automatically.

What is prime factorisation?

Prime factorisation expresses a number as a product of prime numbers. For example, 12 = 2 × 2 × 3 = 2² × 3.

Can I use negative numbers?

Factors are typically positive for elementary work. Our calculator only accepts positive integers.