Updated Number Theory & Algebra Tool

Modular Arithmetic Calculator

Compute remainders, modular operations, modular exponentiation, inverses, greatest common divisors, solve linear congruences, Chinese Remainder Theorem systems and Euler’s totient with exact integer arithmetic.

Remainders & Operations Modular Powers & Inverses GCD & Linear Congruences CRT & Totient Function

Advanced Modular Arithmetic & Number Theory Suite

This Modular Arithmetic Calculator is built for algebra, number theory and cryptography work. It uses exact integer arithmetic to compute remainders, modular operations, powers, inverses, greatest common divisors, linear congruence solutions, Chinese Remainder Theorem combinations and Euler’s totient for suitable inputs.

Modular arithmetic studies remainders when dividing by a positive integer \(n\). For integers \(a\) and \(n>0\), write \[ a = qn + r,\quad 0 \le r < n, \] where \(q\) is the quotient and \(r\) is the remainder. In congruence notation, \[ a \equiv r \pmod{n}. \]

In modular arithmetic, the operations are defined by first performing the usual integer operation and then reducing the result modulo \(n\): \[ (a + b) \bmod n,\quad (a - b) \bmod n,\quad (a \cdot b) \bmod n. \] This tab computes all three for the same inputs.

Modular exponentiation computes powers in modular arithmetic using \[ a^b \bmod n, \] typically via repeated squaring so that very large exponents can be handled efficiently.

An integer \(a\) has a modular inverse modulo \(n\) if there exists an integer \(x\) such that \[ ax \equiv 1 \pmod{n}. \] This happens exactly when \(\gcd(a,n) = 1\). The inverse can be found using the extended Euclidean algorithm.

The greatest common divisor of integers \(a\) and \(b\) is denoted \(\gcd(a,b)\). The extended Euclidean algorithm finds integers \(x\) and \(y\) such that \[ ax + by = \gcd(a,b), \] which is fundamental for modular inverses and linear congruences.

A linear congruence \[ ax \equiv b \pmod{n} \] has a solution if and only if \(d = \gcd(a,n)\) divides \(b\). When solutions exist, there are exactly \(d\) incongruent solutions modulo \(n\), forming a full residue class modulo \(n/d\).

The Chinese Remainder Theorem solves systems of congruences \[ x \equiv a_i \pmod{n_i} \] when the moduli \(n_1,\dots,n_k\) are pairwise coprime. There is then a unique solution modulo \(N = n_1 n_2 \dots n_k\).

Euler’s totient function \(\varphi(n)\) counts the integers between 1 and \(n\) that are coprime to \(n\). If \[ n = p_1^{k_1} p_2^{k_2}\dots p_r^{k_r} \] is the prime factorization of \(n\), then \[ \varphi(n) = n \prod_{i=1}^r \left(1 - \frac{1}{p_i}\right). \] This tab factors \(n\) by trial division for moderate values.

A complete residue system modulo \(n\) contains exactly one representative of each congruence class modulo \(n\), often taken as \[ 0,1,2,\dots,n-1. \] The reduced residue system consists of all integers between 1 and \(n-1\) that are coprime to \(n\).

Modular Arithmetic Calculator – Remainders, Inverses and Congruences

The Modular Arithmetic Calculator on MyTimeCalculator brings together core tools from number theory in one place. You can compute remainders, perform modular addition, subtraction and multiplication, evaluate modular powers, find modular inverses, compute greatest common divisors, solve linear congruences, apply the Chinese Remainder Theorem and explore Euler’s totient and residue systems.

Modular arithmetic underpins many modern applications, from cryptographic protocols and hash functions to coding theory, random number generators and algebraic structures like rings and finite fields.

1. Congruences and Remainders

Fix a positive integer \(n\), called the modulus. For any integer \(a\), there exist unique integers \(q\) and \(r\) such that

\[ a = qn + r,\quad 0 \le r < n. \]

The integer \(r\) is called the remainder of \(a\) modulo \(n\) and is written \(a \bmod n = r\). Two integers \(a\) and \(b\) are congruent modulo \(n\) if they leave the same remainder upon division by \(n\), written

\[ a \equiv b \pmod{n}. \]

This means that \(n\) divides the difference:

\[ a \equiv b \pmod{n} \quad\Longleftrightarrow\quad n \mid (a - b). \]

2. Modular Operations

Once a modulus \(n\) is fixed, arithmetic operations are performed in the usual way and then reduced modulo \(n\). For integers \(a\), \(b\) and modulus \(n\) we define:

\[ (a + b) \bmod n,\quad (a - b) \bmod n,\quad (a \cdot b) \bmod n. \]

These operations respect congruence: if \(a \equiv a'\pmod{n}\) and \(b \equiv b'\pmod{n}\), then

\[ a + b \equiv a' + b' \pmod{n},\quad a - b \equiv a' - b' \pmod{n},\quad ab \equiv a'b' \pmod{n}. \]

This stability under operations is what makes congruence classes modulo \(n\) form a ring.

3. Modular Exponentiation

Powers like \(a^b \bmod n\) appear in many contexts, including public-key cryptography. Computing powers by repeated multiplication is too slow for large exponents, so an exponentiation by squaring method is used. The calculator implements the standard fast method, which repeatedly uses identities such as

\[ a^{2k} = \bigl(a^k\bigr)^2,\quad a^{2k+1} = a \cdot a^{2k}. \]

At each step, intermediate results are reduced modulo \(n\), keeping the numbers small and computations fast.

4. Greatest Common Divisors and Inverses

The greatest common divisor \(\gcd(a,b)\) of two integers \(a\) and \(b\) is the largest integer that divides both. The Euclidean algorithm computes \(\gcd(a,b)\) by repeated division. The extended Euclidean algorithm goes further and finds integers \(x\) and \(y\) such that

\[ ax + by = \gcd(a,b). \]

When \(\gcd(a,n) = 1\), there exists an integer \(x\) such that

\[ ax \equiv 1 \pmod{n}, \]

and the extended algorithm provides such an \(x\). This \(x\) is called the modular inverse of \(a\) modulo \(n\) and is crucial in solving congruences and in cryptographic protocols like RSA.

5. Linear Congruences

Consider a linear congruence of the form

\[ ax \equiv b \pmod{n}. \]

Let \(d = \gcd(a,n)\). The congruence has:

  • No solution if \(d\) does not divide \(b\).
  • Exactly \(d\) incongruent solutions modulo \(n\) if \(d\) divides \(b\).

When \(d \mid b\), one usually divides the entire congruence by \(d\) to obtain

\[ a'x \equiv b' \pmod{n'}, \]

where \(a' = a/d\), \(b' = b/d\) and \(n' = n/d\). Now \(\gcd(a',n') = 1\), so \(a'\) has a modular inverse modulo \(n'\). This leads to a particular solution \(x_0\), and all solutions are given by

\[ x \equiv x_0 + k n' \pmod{n},\quad k \in \mathbb{Z}. \]

6. Chinese Remainder Theorem

The Chinese Remainder Theorem concerns systems of simultaneous congruences

\[ x \equiv a_1 \pmod{n_1},\quad x \equiv a_2 \pmod{n_2},\quad \dots,\quad x \equiv a_k \pmod{n_k}, \]

where the moduli \(n_1,\dots,n_k\) are pairwise coprime. If this condition holds, there exists a unique solution modulo \(N = n_1 n_2 \dots n_k\). One standard construction sets

\[ N_i = \frac{N}{n_i},\quad M_i \equiv N_i^{-1} \pmod{n_i}, \]

and then defines

\[ x \equiv \sum_{i=1}^k a_i M_i N_i \pmod{N}. \]

The CRT tab in the calculator automates these steps, checking coprimality and combining the congruences into a single solution modulo \(N\).

7. Euler’s Totient and Residue Systems

Euler’s totient function \(\varphi(n)\) counts the integers between 1 and \(n\) that are coprime to \(n\). If \(n\) factors as

\[ n = p_1^{k_1} p_2^{k_2}\dots p_r^{k_r}, \]

then

\[ \varphi(n) = n \prod_{i=1}^r \left(1 - \frac{1}{p_i}\right) = \prod_{i=1}^r p_i^{k_i - 1}(p_i - 1). \]

The reduced residue system modulo \(n\) consists of all integers in \(\{1,2,\dots,n-1\}\) that are coprime to \(n\), and its size is precisely \(\varphi(n)\). The calculator factors \(n\) for moderate sizes and displays both the factorization and the totient value.

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Modular Arithmetic Calculator FAQs

Frequently Asked Questions

Quick answers to common questions about remainders, inverses, linear congruences, CRT systems and how to use this modular arithmetic calculator effectively.

The notation \(a \equiv b \pmod{n}\) means that \(a\) and \(b\) leave the same remainder when divided by \(n\). Equivalently, their difference \(a - b\) is divisible by \(n\). For example, \(17 \equiv 5 \pmod{12}\) because both 17 and 5 leave remainder 5 on division by 12, and \(17 - 5 = 12\) is a multiple of 12.

An integer \(a\) has an inverse modulo \(n\) if and only if \(a\) and \(n\) are coprime, meaning \(\gcd(a,n) = 1\). If the greatest common divisor is larger than 1, the equation \(ax \equiv 1 \pmod{n}\) has no solution. The calculator checks this condition using the extended Euclidean algorithm and reports whether an inverse exists, and if so, what its value is.

For a congruence \(ax \equiv b \pmod{n}\), the greatest common divisor \(d = \gcd(a,n)\) measures how strongly \(a\) and \(n\) share common factors. The congruence can only be satisfied if \(b\) is divisible by the same \(d\); otherwise there is no way to reconcile both sides modulo \(n\). When \(d\) divides \(b\), the congruence reduces to one where the coefficient of \(x\) is invertible modulo a smaller modulus, and all solutions can be described in a single formula.

The classical form of the Chinese Remainder Theorem assumes that all moduli are pairwise coprime so that a unique solution exists modulo the product \(N\). If two moduli share a factor, the system can become inconsistent or have more complicated solution sets. This calculator focuses on the standard case and therefore requires pairwise coprime moduli, which guarantees both existence and uniqueness of the solution modulo \(N\).

Computing Euler’s totient function requires factoring \(n\) into primes. Factoring very large integers is computationally expensive, and for many cryptographic sizes it is intentionally hard. To keep the tool responsive in a web browser, the calculator restricts the totient tab to moderate values of \(n\), where trial division is still practical and safe to run interactively.

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