Number Theory Calculators

Number Theory Calculators - Primes, Factors, and Multiples

Number theory is the branch of mathematics that studies the properties of integers - divisibility, prime numbers, factors, multiples, and the relationships between them. These calculators make number theory accessible: from simplifying fractions using GCF to finding scheduling cycles with LCM.

LCM and GCF Calculator - Calculate the Least Common Multiple and Greatest Common Factor of 2 to 6 numbers simultaneously. Shows prime factorization of each number and step-by-step Euclidean algorithm working for GCF. Essential for simplifying fractions, finding least common denominators, and solving scheduling problems.

Fibonacci Calculator - Find any Fibonacci number F(n) up to F(1000) with exact digit count, check if any number is a Fibonacci number, or generate a sequence between two indices. Shows the golden ratio approximation F(n)/F(n−1) converging to φ ≈ 1.618.

Digital Root Calculator - Find the digital root of any integer by repeatedly summing its digits until a single digit remains. Shows each summation step, computes additive persistence and digital sum, and supports any base from 2 to 36. Also generates digital root tables for integer ranges.

Number Base Converter - Convert integers and fractions between any two bases from 2 to 36: binary, octal, decimal, hexadecimal, and beyond. Shows binary in nibble groups, displays all four common bases simultaneously, and provides full step-by-step positional expansion and repeated-division working.

Perfect Number Checker - Classify any whole number as perfect, abundant, or deficient by summing its proper divisors (the aliquot sum). A number is perfect when its proper divisors sum to itself, like 6 (1 + 2 + 3) and 28 (1 + 2 + 4 + 7 + 14). Lists every divisor and shows the comparison, a classic exercise in divisibility and number classification.

Modular Arithmetic Calculator - Compute (a op b) mod m for addition, subtraction, multiplication, exponentiation, and division. Modular exponentiation uses fast repeated squaring with exact BigInt arithmetic, so results stay correct even for huge exponents like 7^1000 mod 13. Division uses the modular inverse via the extended Euclidean algorithm.

Armstrong Number Checker - Check whether a number equals the sum of its own digits, each raised to the power of the digit count, like 153 = 1³ + 5³ + 3³. Shows the full digit^power expansion, and a List in Range mode finds every Armstrong (narcissistic) number in a chosen span.

Applications of Number Theory

GCF (Greatest Common Factor): Used to simplify fractions (divide numerator and denominator by their GCF), divide items into the largest equal groups, and reduce measurements. Example: 24 apples and 36 oranges can be arranged into GCF(24,36) = 12 identical baskets.

LCM (Least Common Multiple): Used to add fractions with different denominators (find the LCD), schedule repeating events (when will they next coincide?), and solve tiling and packing problems. Example: if bus A runs every 12 minutes and bus B every 18 minutes, they next depart together after LCM(12,18) = 36 minutes.