Modular Arithmetic & Multiplicative Inverse
Calculate exact modular sums, products, powers and inverses, with extended Euclidean steps and an RSA learning example.
About this modular arithmetic & multiplicative inverse
Calculate exact modular sums, products, powers and inverses, with extended Euclidean steps and an RSA learning example Computes exact algebraic and numerical metrics directly in your browser using IEEE 754 precision without round-trip latency.
Formula & calculation mechanics
Evaluates high-precision arithmetic, number theory algorithms, and floating-point representations. Arbitrary-precision calculations utilize JavaScript BigInt to compute factorials, combinations ($nCr$), permutations ($nPr$), greatest common divisor (Euclidean algorithm GCD), least common multiple (LCM), and prime factorizations without 64-bit integer overflow. The IEEE 754 visualizer breaks down 32-bit single-precision and 64-bit double-precision floats into Sign (1 bit), Biased Exponent (8/11 bits), and Mantissa/Significand (23/52 bits), modeling denormal numbers, infinities, and NaNs.
Worked numerical calculation
Scenario: RSA: derive d step by step
Sample input:
Processing: For inverse, follow every division and Bézout identity; use the RSA example to derive d.
Illustrative output:
Domain range & precision boundaries
Enter decimal integers, optionally with +/− and surrounding whitespace. Up to 256 digits per field, counting leading zeros, and 260 UTF-16 code units including whitespace/sign. No decimals, fractions, e notation, hex or thousands separators. m must be positive. a and addition/multiplication b may be negative; power requires b ≥ 0. Inverse ignores b. An inverse exists exactly when gcd(a,m)=1. For m=1, this tool uses the zero-ring convention: all residues and the inverse are 0. The zero exponent, including 0^0, gives 1 mod m. Results use exact integers without floating-point rounding. Limits bound browser work, not file size; there is no file input or download. Very long Euclidean derivations display all steps. Runtime depends on input and device; no universal millisecond guarantee. Native BigInt calculation runs locally, including offline after loading. No added calculation dependency, external service, fee or usage quota.
Calculations in Modular Arithmetic & Multiplicative Inverse use standard IEEE 754 precision or exact rational models. Verify significant accounting or tax figures with authoritative references.
In-browser calculation boundary
Formula calculations and numeric evaluations in Modular Arithmetic & Multiplicative Inverse run synchronously in the browser JavaScript engine with zero server telemetry.
Mathematical formulas & precision standards
Mathematical constants, IEEE 754 floating-point standards, and numerical analysis specifications:
Modular Arithmetic & Multiplicative Inverse verification · Audit standards
How to use it
- Choose addition, multiplication, power or inverse; enter a, m and b when shown.
- Calculate the exact result and inspect the normalized remainder.
- Click the action button or observe immediate live calculations rendered by Modular Arithmetic & Multiplicative Inverse.
Frequently asked questions
Which operations are supported by Modular Arithmetic & Multiplicative Inverse?+
Operational requirements for "Which operations are supported by Modular Arithmetic & Multiplicative Inverse?" are verified in local memory during execution of Modular Arithmetic & Multiplicative Inverse.
When does the inverse exist?+
m must be positive. Inverse ignores b.
What are the input and runtime limits?+
Runtime handling of "What are the input and runtime limits?" executes synchronously in your browser tab without remote delays.
How can I derive RSA d?+
This small textbook example starts with the known primes 61 and 53. Choose e coprime to φ(n), then solve ed ≡ 1 (mod φ(n)); loading the example and calculating displays every Euclidean step for d. This is arithmetic education, not a secure key generator: no prime generation, factoring, padding or constant-time cryptography. e=17, m=3120 → d=2753; 1=(−367)×17+2×3120.