Factor Calculator
Perfect Numbers & Special Cases
| Number | Factors | Type |
|---|---|---|
| 6 | 1, 2, 3, 6 | Perfect Number |
| 12 | 1, 2, 3, 4, 6, 12 | Highly Composite |
| 28 | 1, 2, 4, 7, 14, 28 | Perfect Number |
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 | Perfect Square |
| 60 | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 | Highly Composite |
What are Factors?
Factors are numbers that divide evenly into another number with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
Related Calculators
Understanding Factors
What is a Factor?
A factor of a number is an integer that divides that number exactly (with no remainder). Every positive integer has at least two factors: 1 and itself. Numbers with exactly two factors are called prime numbers.
Definition
If a ÷ b = c with no remainder, where a, b, and c are integers, then b is a factor of a.
Finding Factors
To find all factors of a number, test each integer from 1 up to the square root of that number. Each factor you find comes with a pair.
Example: Factors of 36
36 ÷ 1 = 36 ✓ → factor pair: (1, 36)
36 ÷ 2 = 18 ✓ → factor pair: (2, 18)
36 ÷ 3 = 12 ✓ → factor pair: (3, 12)
36 ÷ 4 = 9 ✓ → factor pair: (4, 9)
36 ÷ 5 = 7.2 ✗
36 ÷ 6 = 6 ✓ → factor pair: (6, 6)
Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factor Pairs
Factors come in pairs that multiply to give the original number. For example, the factor pairs of 24 are (1, 24), (2, 12), (3, 8), and (4, 6).
Prime Factorization
Prime factorization breaks a number down into its prime factors - prime numbers that multiply together to give the original number. This is unique for every number (Fundamental Theorem of Arithmetic).
Example: Prime Factorization of 120
60 = 2 × 30
30 = 2 × 15
15 = 3 × 5
120 = 2³ × 3 × 5 = 2 × 2 × 2 × 3 × 5
Types of Numbers by Factors
Prime Numbers
Numbers with exactly 2 factors (1 and itself).
Composite Numbers
Numbers with more than 2 factors.
Perfect Numbers
Sum of proper factors equals the number.
Highly Composite
More factors than any smaller number.
Counting Factors Formula
If a number n = p₁^a × p₂^b × p₃^c..., where p₁, p₂, p₃ are prime factors, then the total number of factors is (a+1)(b+1)(c+1)...
Example: How many factors does 360 have?
Number of factors = (3+1)(2+1)(1+1) = 4 × 3 × 2 = 24 factors
Applications of Factors
Frequently Asked Questions
What is a factor of a number?
A factor is a number that divides evenly into another number with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each divides 12 exactly. Every number has at least two factors: 1 and itself.
What is prime factorization?
Prime factorization breaks a number into the product of prime numbers. For example, 60 = 2 x 2 x 3 x 5 = 2² x 3 x 5. Every composite number has a unique prime factorization (Fundamental Theorem of Arithmetic). Prime factorization is useful for finding GCF, LCM, and simplifying fractions.
How do you find all factors of a number quickly?
Test divisibility starting from 1 up to the square root of the number. Each factor you find gives you a pair. For 36: test 1-6 (since √36=6). You find pairs (1,36), (2,18), (3,12), (4,9), (6,6). This method works because factors come in pairs that multiply to the original number.
What is the difference between factors and multiples?
Factors divide INTO a number; multiples are numbers you get BY multiplying. Factors of 12: 1, 2, 3, 4, 6, 12 (all divide 12). Multiples of 12: 12, 24, 36, 48... (12 times 1, 2, 3, 4...). A number has finite factors but infinite multiples.