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Calculadora de módulo

Calculate a mod n under the floored, truncated and Euclidean conventions, modular powers a^b mod m of large numbers, and modular inverses, with steps.

Atualizado Exemplos verificados: 10

Integers, decimals or fractions.
Experimentar
Resultado
Resultado: 3
Máximo de casas decimais: 12; Ao mais próximo; empates afastando-se de zero
Truncated remainder (sign of a)
−2
Euclidean remainder (never negative)
3
Floored quotient ⌊a ÷ n⌋
−4

−17 ÷ 5 leaves 3 when the quotient is rounded down to −4 (Python, Excel MOD), but −2 when it is rounded toward zero to −3 (C, JavaScript %).

Where a sits between multiples of n

−20−15a = −17floored r = 3truncated r = −2
Como é calculado S

  1. Divide

    an=−175=−175≈−3.4\frac{a}{n} = \frac{-17}{5} = -\frac{17}{5} \approx -3.4

  2. Floored remainder (sign of the divisor)

    q=⌊−175⌋=−4,r=a−nq=−17−5⋅(−4)=3q = \left\lfloor -\frac{17}{5} \right\rfloor = -4,\quad r = a - n q = -17 - 5 \cdot \left(-4\right) = 3

    Python's %, Excel's MOD and most maths texts round the quotient down.

  3. Truncated remainder (sign of the dividend)

    q=trunc⁡(−175)=−3,r=−17−5⋅(−3)=−2q = \operatorname{trunc}\left(-\frac{17}{5}\right) = -3,\quad r = -17 - 5 \cdot \left(-3\right) = -2

    C, Java and JavaScript's % round the quotient toward zero.

  4. Euclidean remainder (never negative)

    r=a−∣n∣⌊a∣n∣⌋=3r = a - |n| \left\lfloor \frac{a}{|n|} \right\rfloor = 3

Sobre Calculadora de módulo

The modulo operation a mod n gives the remainder left when a is divided by n. For positive numbers every convention agrees (17 mod 5 = 2), but for negative numbers they split: the floored remainder r = a − n⌊a/n⌋ takes the sign of the divisor, the truncated remainder rounds the quotient toward zero and takes the sign of the dividend, and the Euclidean remainder is never negative. The calculator shows all three, and also computes modular powers by square-and-multiply and modular inverses by the extended Euclidean algorithm.

Clock and calendar arithmetic, hashing and cyclic buffers all use remainders, and cryptography rests on modular powers. The default, −17 mod 5, is 3 under floored division (Python's %, Excel's MOD) but −2 under truncated division (the % of C and JavaScript). The textbook RSA example encrypts 65 as 65^17 mod 3233 = 2790.

Modular powers stay exact for exponents as large as 10^18 because every squaring is reduced mod m. An inverse a⁻¹ mod m exists only when gcd(a, m) = 1.

Exemplos resolvidos

−17 mod 5

Calculate
Remainder a mod n
Dividend a
-17
Divisor n
5
Resultado
3
Truncated remainder (sign of a)
-2
Euclidean remainder (never negative)
3
Floored quotient ⌊a ÷ n⌋
-4

Fonte de verificação: Python 3.8: -17 % 5 = 3, math.fmod(-17, 5) = -2.0, -17 // 5 = -4

17 mod −5

Calculate
Remainder a mod n
Dividend a
17
Divisor n
-5
Resultado
-3
Truncated remainder (sign of a)
2
Euclidean remainder (never negative)
2

Fonte de verificação: Python 3.8: 17 % -5 = -3, math.fmod(17, -5) = 2.0; Euclidean 17 − 5·⌊17/5⌋ = 2

7.5 mod 2

Calculate
Remainder a mod n
Dividend a
7.5
Divisor n
2
Resultado
1.5
Truncated remainder (sign of a)
1.5

Fonte de verificação: Python 3.8: 7.5 % 2 = 1.5

4^13 mod 497

Calculate
Power a^b mod m
Base a
4
Exponent b
13
Modulus m
497
Resultado
445

Fonte de verificação: Wikipedia — Modular exponentiation worked example; Python pow(4, 13, 497) = 445

Perguntas

How do you calculate a mod n?

Divide, round the quotient down, and subtract: a mod n = a − n⌊a/n⌋. For 17 mod 5, 17 ÷ 5 = 3.4, which rounds down to 3, and 17 − 5 × 3 = 2. For −17 mod 5, −3.4 rounds down to −4, and −17 − 5 × (−4) = 3. On a 12-hour clock, 15:00 is 15 mod 12 = 3 o'clock.

Why do Python and JavaScript give different answers for a negative modulo?

They round the quotient differently. Python's % floors it, so −17 % 5 = 3, with the sign of the divisor; JavaScript, C and Java truncate toward zero, so −17 % 5 = −2, with the sign of the dividend. Both satisfy a = n × q + r. Excel's MOD matches Python. In JavaScript, ((a % n) + n) % n gives the floored answer when n is positive.

How do you calculate large powers modulo a number?

Use square-and-multiply: write the exponent in binary, square repeatedly, and reduce mod m after every step so the numbers never grow past m². For 4^13 mod 497, 13 is 1101 in binary and the answer is 445, the same as Python's pow(4, 13, 497). Computing 4^13 = 67,108,864 first works here, but not for exponents like 10^18.

What is a modular inverse?

The inverse of a modulo m is the number x with a × x ≡ 1 (mod m). 3⁻¹ mod 11 = 4 because 3 × 4 = 12 = 11 + 1. It exists only when gcd(a, m) = 1, so 2 has no inverse mod 10. The extended Euclidean algorithm finds it; in the textbook RSA example the private key 2753 is the inverse of 17 mod 3120, since 17 × 2753 = 46,801 = 15 × 3120 + 1.

What is the difference between remainder and modulo?

For positive numbers they agree: 17 divided by 5 leaves 2 either way. For negative numbers the remainder in the C and JavaScript sense follows the sign of the dividend (−17 rem 5 = −2), while modulo in the mathematical sense follows the divisor or is never negative (−17 mod 5 = 3). When the two differ, they differ by exactly |n|.

Qual é a precisão de “Calculadora de módulo”?

A precisão depende dos dados inseridos e das hipóteses do método. O cálculo decimal usa 50 algarismos significativos, mas estimativas, métodos numéricos e dados de origem podem ter menor precisão; o arredondamento exibido não elimina essas limitações. Exemplos resolvidos verificados com fontes independentes: 10. Por exemplo, “−17 mod 5” é verificado com Python 3.8: -17 % 5 = 3, math.fmod(-17, 5) = -2.0, -17 // 5 = -4.

De onde vem o método?

Knuth, The Art of Computer Programming Vol. 1, §1.2.4 (mod) and Vol. 2, §4.6.3 (powers); Leijen (2001), Division and modulus for computer scientists; Wikipedia — Modular exponentiation (4^13 mod 497 example); Microsoft Excel MOD function.

Sobre esta calculadora

a mod n=a−n⌊an⌋ab mod m by square-and-multiplya a−1≡1(modm)\begin{gathered} a \bmod n = a - n\left\lfloor \frac{a}{n} \right\rfloor \\[6pt] a^{b} \bmod m \text{ by square-and-multiply} \\[6pt] a\,a^{-1} \equiv 1 \pmod m \end{gathered}

Fontes

  1. Knuth, The Art of Computer Programming Vol. 1, §1.2.4 (mod) and Vol. 2, §4.6.3 (powers)
  2. Leijen (2001), Division and modulus for computer scientists
  3. Wikipedia — Modular exponentiation (4^13 mod 497 example)
  4. Microsoft Excel MOD function

Verificado com as referências

Esta calculadora inclui 10 exemplos resolvidos com respostas de fontes independentes. Eles fazem parte do conjunto de testes e você também pode executá-los aqui.

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