Number Theory Calculators
Explore the properties and relationships of integers with our free number theory calculators. Find greatest common factors, least common multiples, prime factorizations, and number sequences instantly with detailed step-by-step solutions perfect for math students and educators.
GCF Calculator
Find the greatest common factor of two or more numbers using prime factorization.
LCM Calculator
Calculate the least common multiple of two or more numbers with step-by-step methods.
Factor Calculator
Find all factors of a number and determine if a number is prime or composite.
Prime Factorization Calculator
Break down any number into its prime factors with a factor tree visualization.
Common Factor Calculator
Find all common factors shared by two or more numbers with clear factor lists.
Number Sequence Calculator
Identify patterns in number sequences and calculate the next terms in the series.
How to Choose the Right Number Theory Calculator
Use the GCF calculator when you need to find the largest number that divides evenly into two or more numbers. This is essential for simplifying fractions (divide both numerator and denominator by their GCF), finding common denominators, and solving problems involving shared grouping or distribution.
The LCM calculator finds the smallest number that is a multiple of two or more numbers. It is the go-to tool for finding common denominators when adding or subtracting fractions, scheduling problems (when will two events coincide?), and determining repeating patterns.
Choose the factor calculator when you need a complete list of all numbers that divide evenly into a given number. It is useful for divisibility analysis, number classification, and understanding the structure of integers. The prime factorization calculator takes this further by breaking a number down into its prime building blocks.
The common factor calculator is designed specifically for comparing the factors of multiple numbers side by side, making it easy to visualize what numbers share in common. The number sequence calculator helps you identify patterns in sequences and predict future terms.
Understanding Number Theory
Number theory is the branch of mathematics devoted to the study of integers and their properties. Often called "the queen of mathematics" by Carl Friedrich Gauss, it explores fundamental questions about divisibility, prime numbers, and the relationships between whole numbers that underpin modern cryptography, computer science, and abstract algebra.
Prime numbers are the building blocks of all integers. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be uniquely represented as a product of prime numbers (its prime factorization). This uniqueness is the foundation for many results in mathematics and is crucial to encryption algorithms like RSA.
The greatest common factor (GCF) and least common multiple (LCM) are two fundamental concepts that describe relationships between numbers. The GCF of two numbers is the largest number that divides both of them, while the LCM is the smallest number that both divide into. These two values are connected by the formula: GCF(a, b) x LCM(a, b) = a x b. The Euclidean algorithm, dating back over 2,300 years, provides an efficient method for computing the GCF.
Number sequences reveal patterns in mathematics that often have deep theoretical significance. Arithmetic sequences have a constant difference between consecutive terms (2, 5, 8, 11, ...), geometric sequences have a constant ratio (3, 6, 12, 24, ...), and the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, ...) where each term is the sum of the two preceding terms appears throughout nature, art, and computer science.
Frequently Asked Questions
What is the difference between GCF and LCM?
The GCF (Greatest Common Factor) is the largest number that divides evenly into all given numbers. The LCM (Least Common Multiple) is the smallest number that all given numbers divide into evenly. For example, for 12 and 18: GCF = 6 (largest shared factor) and LCM = 36 (smallest shared multiple). GCF is used to simplify fractions, while LCM is used to find common denominators.
How do I find the prime factorization of a number?
Start by dividing the number by the smallest prime (2). Keep dividing by 2 until you can no longer do so evenly, then move to the next prime (3), then 5, 7, 11, and so on. Continue until the quotient is 1. For example, 360 = 2 x 2 x 2 x 3 x 3 x 5 = 2^3 x 3^2 x 5. Our prime factorization calculator shows this process step by step using a factor tree.
How do I identify the pattern in a number sequence?
Look at the differences between consecutive terms. If the differences are constant, it is an arithmetic sequence. If the ratios between consecutive terms are constant, it is a geometric sequence. If each term is the sum of the two before it, it is a Fibonacci-type sequence. For more complex patterns, look at second differences (differences of differences) or consider polynomial, exponential, or recursive formulas.
What is a prime number and why does it matter?
A prime number is a whole number greater than 1 whose only factors are 1 and itself (2, 3, 5, 7, 11, 13, ...). Primes matter because every whole number can be uniquely expressed as a product of primes (Fundamental Theorem of Arithmetic). This property is the basis of modern encryption (RSA algorithm), hash functions, and error-correcting codes that secure online banking, communications, and data storage.
How do I use GCF to simplify a fraction?
Find the GCF of the numerator and denominator, then divide both by that GCF. For example, to simplify 24/36: find GCF(24, 36) = 12, then divide both: 24/12 = 2 and 36/12 = 3, giving the simplified fraction 2/3. Our GCF calculator can compute this directly, and many of our fraction tools apply this simplification automatically.