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Number Factorization Calculator

This factorization calculator can find all factors and factor pairs of any positive or negative integer. Enter an integer and instantly get its factorization results.

Complete Factor Decomposition

Calculate and display all possible factors, including positive and negative numbers, supporting large number operations

Detailed Calculation Steps

Show complete calculation process and mathematical explanations to help understand the factorization principles

Instant Results

Get factorization results immediately, support copying and sharing calculation results

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Enter your number and get instant factorization results

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What are Factors?

Understanding the fundamentals of number factors

Factors are integers that divide another number evenly (with no remainder). For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each of these numbers divides 12 exactly.

Understanding factors is fundamental in mathematics and is essential for solving many algebraic problems, simplifying fractions, and finding greatest common divisors.

Key Properties

Every number has at least two factors: 1 and itself
Factors always come in pairs
The number of factors depends on the number's composition
Example: Factors of 18
1 × 18=18
2 × 9=18
3 × 6=18

Factors: 1, 2, 3, 6, 9, 18

How to Find All Factors

Efficient methods for factor discovery

Step-by-Step Method

1

Find the square root of the number and round down to the nearest integer

2

Test division by all integers from 1 up to this square root

3

For each divisor found, record both the divisor and its quotient as factors

4

Arrange all factors in ascending order

Example: Finding factors of 36

√36 = 6, so test numbers 1 through 6:

36 ÷ 1 = 36 ✓ → factors: 1, 36

36 ÷ 2 = 18 ✓ → factors: 2, 18

36 ÷ 3 = 12 ✓ → factors: 3, 12

36 ÷ 4 = 9 ✓ → factors: 4, 9

36 ÷ 6 = 6 ✓ → factor: 6

All factors: 1, 2, 3, 4, 6, 9, 12, 18, 36

Number Factorization Examples

Practice with real examples to master the concepts

Here are practical examples that demonstrate different factorization scenarios:

Example 1: Small Number

✓

Number: 24

Factors: 1, 2, 3, 4, 6, 8, 12, 24

Method: Test divisors 1-5 (√24 ≈ 4.9)

24÷1=24, 24÷2=12, 24÷3=8, 24÷4=6

Example 2: Perfect Square

i

Number: 49

Factors: 1, 7, 49

Note: Perfect squares have odd number of factors

√49 = 7, so 7 appears once (not twice)

Example 3: Prime Number

P

Number: 37

Factors: 1, 37

Note: Prime numbers have exactly two factors

No divisors found between 2 and √37 ≈ 6.08

Example 4: Large Number

∞

Number: 120

Factors: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

Count: 16 factors total

Highly composite number with many factors

Frequently Asked Questions

Common questions about number factorization

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