What are factors of a number?

Factors (or divisors) are whole numbers that divide evenly into another number without leaving a remainder. When you multiply two factors together, you get the original number. Key properties of factors: (1) Every number has at least two factors: 1 and itself. (2) Factors always come in pairs that multiply to give the original number. (3) The number 1 is a factor of every number. (4) Every number is a factor of itself. Example: Factors of 12: The factors of 12 are 1, 2, 3, 4, 6, and 12 because: 1 × 12 = 12, 2 × 6 = 12, 3 × 4 = 12. Each of these numbers divides 12 evenly: 12 ÷ 1 = 12, 12 ÷ 2 = 6, 12 ÷ 3 = 4, 12 ÷ 4 = 3, 12 ÷ 6 = 2, 12 ÷ 12 = 1. How to find factors: Start with 1 and the number itself. Try dividing by 2, 3, 4, etc., up to the square root. When you find a divisor, both the divisor and the quotient are factors. Our calculator automates this process and shows all factors instantly, along with factor pairs, prime factorization, and special number properties.

What is prime factorization?

Prime factorization is the process of breaking down a composite number into a product of prime numbers. Every composite number has a unique prime factorization (Fundamental Theorem of Arithmetic). Prime factorization format: Number = p₁^a₁ × p₂^a₂ × p₃^a₃ × ... where p₁, p₂, p₃ are prime numbers and a₁, a₂, a₃ are their exponents (how many times each prime appears). Examples: 12 = 2² × 3 = 2 × 2 × 3. The prime factors are 2 and 3. 2 appears twice (exponent 2), 3 appears once (exponent 1). 24 = 2³ × 3 = 2 × 2 × 2 × 3. The prime factors are 2 and 3. 2 appears three times, 3 appears once. 100 = 2² × 5² = 2 × 2 × 5 × 5. The prime factors are 2 and 5, each appearing twice. How to find prime factorization: Divide by the smallest prime (2) as many times as possible. Continue with the next smallest primes (3, 5, 7, 11...). Stop when the remaining number is 1. Why prime factorization is useful: Finding GCF and LCM of multiple numbers. Simplifying fractions and radicals. Solving number theory problems. Understanding divisibility rules. Cryptography (RSA encryption uses prime factorization). Our calculator shows complete prime factorization with exponents, lists all prime factors, and explains the breakdown step by step.

What is the difference between prime and composite numbers?

Prime and composite numbers are two fundamental classifications in number theory. Prime Numbers: Definition: A prime number has exactly 2 factors: 1 and itself. Cannot be divided evenly by any other number. Properties: All primes except 2 are odd (2 is the only even prime). Primes are the building blocks of all numbers. There are infinitely many prime numbers. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47... Composite Numbers: Definition: A composite number has more than 2 factors. Can be divided evenly by numbers other than 1 and itself. Can be broken down into prime factors. Properties: All even numbers except 2 are composite. Has at least 3 factors (1, itself, and at least one other). Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25... Special case - The number 1: 1 is neither prime nor composite. It has only 1 factor (itself). It is called a unit in number theory. Quick identification: Numbers ending in 0, 2, 4, 6, 8 (except 2) are composite. Numbers ending in 5 (except 5) are composite. Use divisibility rules for 3, 7, 11, etc., to test larger numbers. Real-world importance: Primes are used in cryptography and computer security. Composite numbers appear in patterns, arrays, and rectangular arrangements. Prime testing is crucial for random number generation and coding theory. Our calculator instantly identifies whether a number is prime or composite and shows all factors for composite numbers.

What are perfect numbers?

A perfect number is a positive integer that equals the sum of its proper divisors (all factors except the number itself). Perfect numbers are rare and have fascinated mathematicians for over 2,000 years. First perfect numbers: 6: Proper divisors are 1, 2, 3. Sum: 1 + 2 + 3 = 6 ✓. 28: Proper divisors are 1, 2, 4, 7, 14. Sum: 1 + 2 + 4 + 7 + 14 = 28 ✓. 496: Proper divisors sum to 496 ✓. 8128: Proper divisors sum to 8128 ✓. The fifth perfect number is 33,550,336. Related classifications: Perfect: Sum of proper divisors equals the number. Abundant: Sum of proper divisors is greater than the number (e.g., 12: 1+2+3+4+6 = 16 > 12). Deficient: Sum of proper divisors is less than the number (e.g., 8: 1+2+4 = 7 < 8). Properties of perfect numbers: All known perfect numbers are even. Every even perfect number has the form 2^(p-1) × (2^p - 1) where 2^p - 1 is a prime (Mersenne prime). It is unknown whether any odd perfect numbers exist. Only 51 perfect numbers are currently known (as of 2023). The largest known perfect number has over 49 million digits. Historical significance: Known to ancient Greek mathematicians (Euclid, ~300 BCE). Studied by Pythagoreans for mystical properties. Connected to Mersenne primes in modern number theory. Used in computer science for testing algorithms. Our calculator automatically detects perfect numbers and classifies all numbers as perfect, abundant, or deficient.

How do I find factor pairs?

Factor pairs are two numbers that multiply together to give the original number. Every factor has a corresponding pair, and finding factor pairs is a systematic way to identify all factors. Method to find factor pairs: Start with 1 and the number itself (always a factor pair). Test each integer from 2 up to the square root of the number. For each divisor found, pair it with the quotient (number ÷ divisor). Stop when you reach the square root (to avoid duplicates). Example: Factor pairs of 24. Start: 1 × 24 = 24. Test 2: 2 × 12 = 24. Test 3: 3 × 8 = 24. Test 4: 4 × 6 = 24. Stop at √24 ≈ 4.9. Factor pairs: (1, 24), (2, 12), (3, 8), (4, 6). All factors: 1, 2, 3, 4, 6, 8, 12, 24. Special case - Perfect squares: Perfect squares have an odd number of factors because one factor pairs with itself. Example: 36 = 6 × 6. Factor pairs: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6). The factor 6 pairs with itself. Why factor pairs are useful: Factoring quadratic expressions (x² + bx + c). Finding dimensions of rectangles with a given area. Solving problems involving arrays and groups. Understanding divisibility and multiplication. Simplifying algebraic fractions. Optimization problems (maximizing area with fixed perimeter). Our calculator displays all factor pairs in an organized grid, making it easy to see the relationship between factors and understand the structure of the number.

What is the sum of divisors function?

The sum of divisors function, denoted σ(n), is the sum of all positive divisors (factors) of a number n, including 1 and the number itself. This function is important in number theory and has many applications. Sum of divisors notation: σ(n) = sum of all divisors of n. Example: σ(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28. Sum of proper divisors: Often denoted s(n) or σ(n) - n. This excludes the number itself, only summing proper divisors. Example: s(12) = 1 + 2 + 3 + 4 + 6 = 16. Properties and formulas: For a prime p: σ(p) = p + 1 (only factors are 1 and p). For a prime power p^k: σ(p^k) = 1 + p + p² + ... + p^k = (p^(k+1) - 1) / (p - 1). For coprime numbers: σ(mn) = σ(m) × σ(n) (multiplicative function). Example: σ(6) = σ(2 × 3) = σ(2) × σ(3) = (1+2) × (1+3) = 3 × 4 = 12. Applications: Perfect number test: n is perfect if σ(n) = 2n, or equivalently s(n) = n. Abundant number: σ(n) > 2n, or s(n) > n. Deficient number: σ(n) 2 × 12 = 24 → 12 is abundant. σ(8) = 1 + 2 + 4 + 8 = 15 < 2 × 8 = 16 → 8 is deficient. Our calculator computes both σ(n) (sum of all factors) and s(n) (sum of proper divisors), and uses these to classify the number as perfect, abundant, or deficient with clear explanations.

Factor Calculator - Find All Factors & Prime Factorization | AICalculator.pro

About Factor Calculator

Our Factor Calculator is a comprehensive tool for finding all factors, divisors, and prime factorization of any positive integer. Whether you're a student learning about number theory, a teacher preparing math lessons, or anyone working with divisibility and factorization, this calculator provides instant, detailed analysis of any number up to 10 million.

The calculator not only lists all factors but also identifies special number properties like prime numbers, perfect squares, perfect numbers, and provides complete prime factorization with visual representation. It's perfect for homework help, test preparation, and understanding the fundamental structure of numbers.

Understanding Factors

What are Factors?

Factors (also called divisors) are whole numbers that divide evenly into another number without leaving a remainder. When you multiply two factors together, you get the original number.

Example: Factors of 24

Factors: 1, 2, 3, 4, 6, 8, 12, 24
Factor Pairs: (1×24), (2×12), (3×8), (4×6)
Prime Factorization: 2³ × 3 = 2 × 2 × 2 × 3
Factor Count: 8 factors total

Prime Factorization

Every composite number can be expressed as a unique product of prime numbers. This is called prime factorization and is guaranteed by the Fundamental Theorem of Arithmetic.

Prime Factorization Examples

12 = 2² × 3
24 = 2³ × 3
100 = 2² × 5²
144 = 2⁴ × 3²
1000 = 2³ × 5³

Types of Numbers

Prime vs. Composite

🌟 Prime Numbers

Have exactly 2 factors: 1 and itself

Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...

🔢 Composite Numbers

Have more than 2 factors

Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18...

Perfect Numbers

A perfect number equals the sum of its proper divisors (all factors except the number itself). These rare numbers have fascinated mathematicians for over 2,000 years.

First Perfect Numbers

6: 1 + 2 + 3 = 6
28: 1 + 2 + 4 + 7 + 14 = 28
496: Sum of proper divisors = 496
8128: Sum of proper divisors = 8128

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Additional Resources

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Understanding Factors & Prime Numbers

🔢 What are Factors?

🌟 Prime Numbers

💎 Perfect Numbers

About Factor Calculator

Understanding Factors

What are Factors?

Example: Factors of 24

Prime Factorization

Prime Factorization Examples

Types of Numbers

Prime vs. Composite

🌟 Prime Numbers

🔢 Composite Numbers

Perfect Numbers

First Perfect Numbers

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