Number Theory Calculators
This category presents calculators describing integer behaviour and remainder systems. Each tool provides numerical outputs for factors, modular relationships, divisibility patterns and structural properties across whole numbers.
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Congruence Solver Calculator
A linear congruence links an unknown whole number to a remainder pattern that repeats in fixed steps.
Example use: working out which whole numbers match a repeating countdown on a kitchen timer.
Inputs: coefficient, constant term, modulus
Outputs: greatest common divisor, coprime status, reduced modulus, modular inverse, total unique solutions, principal solution, list of unique solutions
Visual: shows how the repeated remainder values spread across the full modular cycle
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Continued Fraction Expansion Calculator
A continued fraction expresses a ratio as a sequence of whole-number steps that refine its accuracy.
Example use: approximating a measured length when only whole-number steps on a ruler are available.
Inputs: numerator, denominator
Outputs: decimal equivalent, continued fraction, number of terms, expansion type, convergent, maximum error bound, legendre criterion
Visual: shows how each successive approximation moves closer to the target value and how the error decreases
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Diophantine Equation Solver
A linear equation with whole-number constraints may have no whole-number solution, a single pattern of solutions, or infinitely many arranged in regular steps.
Example use: checking whether two different packet sizes can combine to reach a specific total count.
Inputs: first coefficient, second coefficient, constant term
Outputs: greatest common divisor, coprimality status, solvability, bezout identity, particular solution, general solution, lattice distance, geometric angle, smallest positive solution for the first variable, minimal norm solution, simplified equation
Visual: shows whole-number points on a grid and highlights one particular solution among them
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Divisors and Multiples Calculator
A whole number can be broken down into its factors and can generate a predictable sequence of multiples.
Example use: listing all the ways to split a group of items evenly among friends.
Inputs: whole number, number of multiples to show
Outputs: classification, parity, prime factorisation, radical value, euler totient, total divisors, proper divisors count, greatest proper divisor, divisors list, unitary divisors, sum of divisors, harmonic mean, abundancy index, abundancy status, aliquot sequence, first ten multiples
Visual: shows how the multiples increase in regular steps
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Euclidean Algorithm Calculator
Two whole numbers share a greatest common divisor that can be uncovered through repeated remainder steps.
Example use: reducing two different cycle lengths to find when they next align.
Inputs: first integer, second integer
Outputs: greatest common divisor, bezout coefficient for the first integer, bezout coefficient for the second integer, bezout identity, modular inverse, general solution, efficiency measure
Visual: shows the sequence of remainders and quotients produced during the reduction process
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Euler's Totient Function Calculator
A whole number has a specific count of smaller numbers that share no common factor with it.
Example use: checking how many possible step sizes on a circular dial avoid repeating early.
Inputs: whole number
Outputs: distinct prime factors, totient value, coprimality density, sum of coprime integers, average order, sparsely totient status, parity note
Visual: shows how the totient value compares with the number itself and how it behaves across nearby values
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Fermat's Little Theorem Calculator
A whole number raised to repeated powers behaves predictably when reduced under a prime modulus.
Example use: checking how a repeated button press cycles through a fixed set of states.
Inputs: whole number, prime modulus
Outputs: modular result, primality of the modulus, coprimality, euler totient, legendre symbol, modular inverse, order under the modulus, congruence relation
Visual: shows how repeated powers cycle through remainder values under the prime modulus
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Modular Arithmetic Calculator
A whole number can be expressed as a quotient and remainder when divided by a fixed modulus, creating a repeating cycle of values.
Example use: determining the position on a rotating dial after many full turns.
Inputs: whole number, modulus
Outputs: integer quotient, remainder, congruence relation, additive inverse, multiplicative inverse, euler totient, multiplicative order, coprime status
Visual: shows the repeating cycle of remainder values and how powers move around the cycle
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Modular Inverse Calculator
A modular inverse exists when a whole number and a modulus share no common factor other than one.
Example use: reversing a repeated shift in a number puzzle to recover the original value.
Inputs: whole number, modulus
Outputs: greatest common divisor, coprime status, euler totient, multiplicative order, jacobi symbol, bezout coefficients, modular inverse
Visual: shows how the congruence steps lead to the value that reverses the multiplication
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Multiplicative Order Calculator
A whole number raised to increasing powers eventually repeats its remainder under a modulus, forming a cycle with a specific length.
Example use: checking how many button presses are needed before a digital counter returns to its starting value.
Inputs: base number, modulus
Outputs: greatest common divisor, euler totient, carmichael function, subgroup index, primitive root status, group generation status, multiplicative order
Visual: shows the sequence of powers under the modulus and the point where the cycle repeats
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Perfect Numbers Calculator
A whole number can be compared with the sum of its proper divisors to determine whether it is perfect, abundant, or deficient.
Example use: checking whether a number of collected items matches the total contributed by smaller groups.
Inputs: whole number
Outputs: proper divisors, aliquot sum, sum of divisors, abundance index, abundance or deficiency, harmonic mean, semi-perfect status, primitive abundant status, power-of-two status, classification
Visual: shows the divisor values and compares their sum with the original number
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Prime Factorisation Calculator
A whole number can be broken into prime factors that reveal its structure and divisor properties.
Example use: splitting a total count of identical items into prime-sized groups for sorting.
Inputs: whole number
Outputs: prime factors, exponential form, total prime factors, classification, number of divisors, sum of divisors, product of divisors, euler totient, perfect number status
Visual: shows how the prime factors and their exponents contribute to the overall structure
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Primitive Root Finder Calculator
A primitive root is a whole number that generates every non-zero remainder under a prime modulus through its powers.
Example use: checking how a repeated step on a circular counter eventually reaches every possible position.
Inputs: prime number
Outputs: euler totient, number of primitive roots, root density, smallest primitive root, average root value, product of roots under the modulus, sum of roots, primitive roots list
Visual: shows how the powers of each primitive root cover all non-zero remainder values
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Quadratic Residue Checker
A quadratic residue is a remainder that can be produced by squaring some whole number under a modulus.
Example use: checking whether a particular remainder can appear when squaring numbers on a small digital counter.
Inputs: whole number, modulus
Outputs: target congruence, quadratic residue status, solution set, legendre symbol, jacobi symbol, multiplicative order, unique residues count
Visual: shows the set of possible squared remainders and how often each one appears
Number Theory Calculators FAQs
They generate numerical outputs describing remainders, factor structures and shared divisors across different whole-number relationships.
Common factors are identified through repeated division, presenting the largest integer that divides both values without leaving a remainder.
Specific calculators generate integer solutions for linear expressions and modular relationships, presenting values that satisfy the required conditions.
Prime numbers support factor breakdowns and remainder calculations, presenting structured relationships for powers, divisors and modular behaviour.