Find all factors, factor pairs, and prime factorization of any number up to 10 billion. Includes GCF/HCF and LCM calculator for two numbers. Free for US and UK math students and teachers.
Factors are the building blocks of numbers. Understanding how to find factors, prime factorizations, greatest common factors (GCF), and lowest common multiples (LCM) is essential in mathematics from primary school through university. This free factor calculator finds all factors, factor pairs, and prime factorization of any whole number, and calculates GCF/HCF and LCM for two numbers instantly.
A factor of a whole number n is any positive integer that divides n exactly (with no remainder). For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 — because each divides 12 evenly.
Every whole number greater than 1 has at least two factors: 1 and itself. Numbers with exactly two factors (1 and themselves) are called prime numbers. Numbers with more than two factors are called composite numbers.
To find all factors of a number n, test every integer from 1 to √n. If i divides n, then both i and n/i are factors.
Example: Find all factors of 36.
A prime number is a whole number greater than 1 that has no positive divisors other than 1 and itself. The first 20 primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71.
Prime factorization expresses a composite number as a product of prime numbers. By the Fundamental Theorem of Arithmetic, every integer greater than 1 has a unique prime factorization.
Example — Prime Factorization of 360 using the Division Method:
A factor tree is a visual method for finding prime factorization. Start with the number and break it into two factors, then continue breaking composite factors into pairs until all branches end in prime numbers. For example, 72 → 8 × 9 → (2×4) × (3×3) → (2×2×2) × (3×3) → 2³ × 3². Factor trees are widely used in UK primary and secondary schools and US elementary/middle school mathematics.
The Greatest Common Factor (GCF) — called the Highest Common Factor (HCF) in the UK — is the largest positive integer that divides all given numbers without remainder.
Method: Prime Factorization
Find the prime factorization of each number, then multiply the common prime factors using the lowest exponent.
Example: GCF(360, 48)
GCF is used to simplify fractions to their lowest terms. For example, 36/48 simplifies to 36÷12 / 48÷12 = 3/4. The GCF of the numerator and denominator (12) is the simplification factor.
The Lowest Common Multiple (LCM) — also called the Least Common Multiple in the US — is the smallest positive integer that is divisible by all given numbers.
Method: Prime Factorization
Find the prime factorization of each number, then multiply all prime factors using the highest exponent.
Example: LCM(360, 48)
There is an elegant relationship: GCF(a,b) × LCM(a,b) = a × b. For our example: 24 × 720 = 17,280 and 360 × 48 = 17,280. ✓
LCM is used when adding or subtracting fractions with different denominators. The common denominator is the LCM of the individual denominators. For example, to calculate 1/4 + 1/6, the LCM of 4 and 6 is 12, so: 3/12 + 2/12 = 5/12.
A perfect number is a positive integer that equals the sum of its proper divisors (all divisors except itself). The first four perfect numbers are:
Perfect numbers were studied by ancient Greeks and are connected to Mersenne primes (primes of the form 2^p − 1). As of 2025, only 51 perfect numbers are known, all even. Whether any odd perfect numbers exist is an unsolved problem in mathematics.
In the United Kingdom, factors are introduced in Key Stage 2 (ages 7–11) as part of the National Curriculum for Mathematics. Students learn factors and multiples, prime numbers, and simple prime factorization. At GCSE level, students work with HCF, LCM, and prime factorization using index notation (e.g. 2³ × 3 × 5²). These topics appear in the Number strand of all GCSE specifications.
In the United States, factors are covered in 4th–6th grade Common Core Mathematics. By Grade 6, students must find factors of numbers and understand prime and composite numbers. GCF and LCM are explicit Grade 6 standards (6.NS.B.4). Prime factorization and its applications appear in middle school and early high school Algebra.
The difficulty of factoring large numbers into their prime factors is the foundation of RSA public-key cryptography, the most widely used encryption algorithm on the internet. RSA security depends on the mathematical fact that multiplying two large primes is easy, but finding the factors of their product — without knowing them in advance — is computationally infeasible for numbers with hundreds of digits. A 2048-bit RSA key involves two primes of about 600 digits each. Breaking such a key would require more time than the age of the universe using current hardware and algorithms. Quantum computers pose a theoretical future threat (via Shor's algorithm) to RSA encryption.
Factors of a number n are all positive integers that divide n exactly with no remainder. For example, factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. Every number has at least two factors: 1 and itself. Numbers with exactly two factors are prime; all others are composite.
Prime factorization expresses a number as a product of its prime number factors. For example, 360 = 2³ × 3² × 5. By the Fundamental Theorem of Arithmetic, every integer greater than 1 has a unique prime factorization (ignoring order). It is found by the division method or factor tree method.
GCF (Greatest Common Factor) and HCF (Highest Common Factor) are the same thing — the largest number that divides all given numbers without remainder. GCF is the standard term in the USA; HCF is the standard term in the UK. Both mean exactly the same mathematical concept.
Use prime factorization: find the prime factorization of each number, then multiply all prime factors using the highest exponent from either number. Alternatively, LCM(a,b) = a × b ÷ GCF(a,b). For example, LCM(12,8): 12=2²×3, 8=2³, LCM=2³×3=24.
A perfect number equals the sum of its proper divisors (all divisors except itself). The first perfect numbers are 6 (1+2+3=6), 28 (1+2+4+7+14=28), and 496. As of 2025, only 51 perfect numbers are known, all even. Whether any odd perfect numbers exist is an open mathematical question.
To simplify a fraction, divide both numerator and denominator by their GCF. For example, 36/48: GCF(36,48)=12, so 36÷12=3 and 48÷12=4, giving 3/4 in lowest terms. This is a key skill in UK KS3/GCSE and US middle school mathematics.
This calculator handles numbers up to 10 billion (10,000,000,000). For most practical purposes — homework, simplifying fractions, finding GCF/LCM — numbers fit well within this range. Cryptographic applications involve numbers with hundreds of digits, which require specialized algorithms far beyond trial division.
RSA encryption multiplies two large prime numbers to create a public key. Security comes from the fact that factoring the product back into its primes is computationally infeasible for numbers with hundreds of digits. A 2048-bit RSA key involves primes of about 600 digits each. Breaking it would take longer than the age of the universe with current computers.
Results are for educational purposes. For very large numbers (near 10 billion), the factorization algorithm may take a moment. JavaScript uses 64-bit floating-point numbers; integers up to 2^53 (about 9 quadrillion) are exact, but this calculator limits input to 10 billion for performance.