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

Factors Finder

Enter a positive integer

Factors of 21: 1, 3, 7, 21

Factor pairs: 1 × 21, 3 × 7

Prime factorisation: 3 × 7

Total factors: 4

Step‑by‑Step

Number: 21

Step 1: Find all pairs of integers whose product is 21.

1 × 21

3 × 7

Step 2: Collect all unique numbers from these pairs.

Factors: 1, 3, 7, 21

Step 3: Prime factorisation (optional): divide 21 by the smallest prime repeatedly.

21 = 3 × 7

Total number of factors: 4

Tip: Factors come in pairs (i × j = n). The number of factors is even unless n is a perfect square.

The Factors Calculator finds all positive divisors (factors) of any integer. It also shows factor pairs, prime factorisation, and the total number of factors. For example, the factors of 21 are 1, 3, 7, 21 (since 21 = 3 × 7). This tool is perfect for algebra, simplifying fractions, finding common divisors, and exploring number theory.

What Are Factors?

A factor (or divisor) of a whole number n is another whole number that divides n exactly with no remainder. Every number has at least two factors: 1 and itself. For 21, the complete set is 1, 3, 7, 21. Notice that 3 and 7 are primes, so the prime factorisation of 21 is 3 × 7.

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 21 items into equal groups – possible into 1, 3, 7, or 21 groups.

Understanding Prime Factorisation

Every integer greater than 1 can be uniquely expressed as a product of prime numbers – the Fundamental Theorem of Arithmetic. For 21, the prime factors are 3 and 7 (21 = 3 × 7). Knowing the prime factorisation helps compute the number of factors: if n = p₁ᵃ × p₂ᵇ × … then the number of factors = (a+1)(b+1)… . For 21 = 3¹ × 7¹, factor count = (1+1)(1+1) = 4, which matches 1,3,7,21.

Factor Table for Common Numbers

Number	Factors
12	1,2,3,4,6,12
15	1,3,5,15
21	1,3,7,21
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

Step‑by‑Step Example: Factors of 21

To find factors of 21: test divisibility from 1 to √21 ≈ 4.58.
– 1 divides 21 → include 1 and 21/1 = 21 → factors 1,21.
– 2 does not divide 21 (21 ÷ 2 = 10.5).
– 3 divides 21 → include 3 and 21/3 = 7 → factors 3,7.
– 4 does not divide 21.
– Stop because 5 greater than √21.
Sorted factors: 1,3,7,21.
Factor pairs: 1×21, 3×7. Prime factorisation: 21 = 3 × 7.

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. Try entering 21 or any other integer to see the results.

Quick Manual Example (21)

Input: 21
Test i=1 → 21÷1=21 → factors 1,21
i=2 → not divisible
i=3 → 21÷3=7 → factors 3,7
i=4 greater than √21 → stop.
Final factors: 1,3,7,21

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 21 are 1, 3, 7, 21.

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, 21 = 3 × 7.

Can I use negative numbers?

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