Find all factors and prime factorization of any positive integer. Perfect for math homework, number theory, and solving problems involving divisibility with detailed step-by-step solutions and factor trees.
A factor of a number is any integer that divides evenly into that number without leaving a remainder. Understanding factors is fundamental to many mathematical concepts including simplifying fractions, finding common denominators, solving equations, and working with divisibility rules.
Prime factorization breaks down a number into its basic building blocks - prime numbers that when multiplied together give the original number. This process is essential for advanced mathematics, cryptography, and computer science applications.
Every positive integer has at least two factors: 1 and itself
Express any number as a product of prime powers
| Number | All Factors | Prime Factorization | Factor Count |
|---|---|---|---|
| 12 | 1, 2, 3, 4, 6, 12 | 2² × 3 | 6 |
| 18 | 1, 2, 3, 6, 9, 18 | 2 × 3² | 6 |
| 24 | 1, 2, 3, 4, 6, 8, 12, 24 | 2³ × 3 | 8 |
| 30 | 1, 2, 3, 5, 6, 10, 15, 30 | 2 × 3 × 5 | 8 |
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 | 2² × 3² | 9 |
| 48 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 | 2⁴ × 3 | 10 |
| 60 | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 | 2² × 3 × 5 | 12 |
Prime Numbers: Numbers with exactly two factors - 1 and themselves. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...
Composite Numbers: Numbers with more than two factors. Can be broken down into prime factors. Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18...
Special Cases: 1 is neither prime nor composite (has only one factor). 2 is the only even prime number.
Factor Pairs: Factors always come in pairs. If a is a factor of n, then n/a is also a factor.
Square Numbers: Perfect squares have an odd number of factors because one factor pairs with itself (√n × √n = n).
Factor Count Formula: If n = p₁^a₁ × p₂^a₂ × p₃^a₃, then the number of factors = (a₁+1)(a₂+1)(a₃+1)...
Divisibility Rules: Use shortcuts to identify factors: divisible by 2 (even), by 3 (sum of digits divisible by 3), by 5 (ends in 0 or 5), etc.
Complete Factor Method:
1. Start with 1 and the number itself
2. Test each integer from 2 to √n for divisibility
3. For each divisor found, include both the divisor and quotient
4. Arrange factors in ascending order
5. Check for completeness (no gaps in factor pairs)
Prime Factorization Method:
1. Start with the smallest prime number (2)
2. Divide the number by this prime as many times as possible
3. Move to the next prime number
4. Repeat until the quotient becomes 1
5. Express as powers of primes
| Application | Scenario | Factor Usage | Example |
|---|---|---|---|
| Fraction Simplification | Reducing fractions | Common factors in numerator/denominator | 24/36 = 2/3 (factor of 12) |
| Array Arrangements | Organizing objects | Possible rectangular arrangements | 24 items: 1×24, 2×12, 3×8, 4×6 |
| Time Planning | Scheduling events | Even divisions of time periods | 60 minutes: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 |
| Cryptography | RSA encryption | Large prime factorization difficulty | Security based on factoring large numbers |
| Music Theory | Rhythm patterns | Beat subdivisions | 12 beats: 1, 2, 3, 4, 6, 12 divisions |
| Engineering | Gear ratios | Mechanical advantage calculations | Gear teeth must be factor-related |
| Computer Science | Algorithm optimization | Loop structure and data organization | Array indexing and partitioning |
Inverse Relationship: If a is a factor of b, then b is a multiple of a. These concepts are mathematically inverse.
Range Properties: All factors of n are ≤ n, while multiples of n are ≥ n (except for negative numbers).
Finite vs Infinite: Every number has a finite number of factors but infinite multiples.
Perfect Numbers: Numbers equal to the sum of their proper factors (excluding the number itself). Example: 6 = 1 + 2 + 3.
Abundant Numbers: Numbers where the sum of proper factors exceeds the number. Example: 12 (1+2+3+4+6 = 16 > 12).
Deficient Numbers: Numbers where the sum of proper factors is less than the number. Example: 8 (1+2+4 = 7 < 8).
Highly Composite Numbers: Numbers with more factors than any smaller positive integer. Examples: 1, 2, 4, 6, 12, 24, 36, 48, 60, 120...
| Method | Best For | Time Complexity | Advantages |
|---|---|---|---|
| Trial Division | Small to medium numbers | O(√n) | Simple, finds all factors |
| Prime Factorization | Numbers with known prime structure | O(√n) | Systematic, reveals structure |
| Sieve Methods | Multiple numbers simultaneously | O(n log log n) | Efficient for ranges |
| Factor Trees | Visual learners, education | O(√n) | Easy to understand and verify |
Fundamental Theorem of Arithmetic: Every integer greater than 1 has a unique prime factorization (except for order of factors).
Divisor Function: τ(n) counts the number of positive divisors of n. For n = p₁^a₁ × p₂^a₂ × ..., then τ(n) = (a₁+1)(a₂+1)...
Euler's Totient Function: φ(n) counts integers from 1 to n that are coprime to n. Related to prime factorization.
Square Root Limit: Only test divisors up to √n since factors come in pairs.
Skip Even Numbers: After testing 2, only test odd numbers for odd inputs.
Use Divisibility Rules: Quick tests for 2, 3, 5, 9, 11 can speed up factor finding.
Start with Small Primes: Begin factorization with 2, 3, 5, 7, 11... for efficiency.
GCD and LCM: Factor finding is essential for calculating greatest common divisors and least common multiples.
Fraction Operations: Simplifying fractions requires finding common factors between numerator and denominator.
Modular Arithmetic: Understanding factors helps in solving congruence equations and modular problems.
Combinatorics: Factor analysis helps in counting problems and arrangement calculations.