# Kalkulator modulo

> 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.

Versi interaktif: https://www.calcopenly.com/id/math/modulo-calculator
Topik: Kalkulator matematika

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.

## Masukan

- **Calculate** (pilihan: Remainder a mod n, Power a^b mod m, Inverse a⁻¹ mod m)
- **Dividend a**: Integers, decimals or fractions.
- **Divisor n**
- **Base a**
- **Exponent b**: Any size, e.g. 10^18. A negative exponent uses the inverse of a.
- **Modulus m**
- **Number a**
- **Modulus m**

## Hasil

- Hasil — hasil utama
- Truncated remainder (sign of a)
- Euclidean remainder (never negative)
- Floored quotient ⌊a ÷ n⌋
- Nilai tepat

## Rumus

$$
\begin{gathered} a \bmod n = a - n\left\lfloor \frac{a}{n} \right\rfloor \\[6pt] a^{b} \bmod m \text{ by square-and-multiply} \\[6pt] a\,a^{-1} \equiv 1 \pmod m \end{gathered}
$$

## Contoh penyelesaian

### −17 mod 5

- Calculate: Remainder a mod n
- Dividend a: -17
- Divisor n: 5
- **Hasil: 3**
- **Truncated remainder (sign of a): -2**
- **Euclidean remainder (never negative): 3**
- **Floored quotient ⌊a ÷ n⌋: -4**
- Sumber pemeriksaan: 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
- **Hasil: -3**
- **Truncated remainder (sign of a): 2**
- **Euclidean remainder (never negative): 2**
- Sumber pemeriksaan: 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
- **Hasil: 1.5**
- **Truncated remainder (sign of a): 1.5**
- Sumber pemeriksaan: 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
- **Hasil: 445**
- Sumber pemeriksaan: Wikipedia — Modular exponentiation worked example; Python pow(4, 13, 497) = 445

### 2^(10^18) mod 1 000 000 007

- Calculate: Power a^b mod m
- Base a: 2
- Exponent b: 10^18
- Modulus m: 1000000007
- **Hasil: 719,476,260**
- Sumber pemeriksaan: Python 3.8 pow(2, 10**18, 10**9 + 7) (scratchpad forkA/verify_mod.py)

### RSA encryption 65^17 mod 3233

- Calculate: Power a^b mod m
- Base a: 65
- Exponent b: 17
- Modulus m: 3233
- **Hasil: 2,790**
- Sumber pemeriksaan: Wikipedia — RSA (cryptosystem) worked example, c = 65^17 mod 3233 = 2790

## Pertanyaan

### 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|.

### Seberapa akurat “Kalkulator modulo”?

Akurasi bergantung pada masukan dan asumsi metode. Aritmetika desimal memakai 50 digit signifikan, tetapi perkiraan, metode numerik dan data sumber bisa kurang presisi; pembulatan yang ditampilkan tidak menghilangkan batasan itu. Contoh penyelesaian yang diperiksa dengan sumber independen: 10. Misalnya, “−17 mod 5” diperiksa dengan Python 3.8: -17 % 5 = 3, math.fmod(-17, 5) = -2.0, -17 // 5 = -4.

### Dari mana metode ini berasal?

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.

## Sumber

- 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](https://www.microsoft.com/en-us/research/publication/division-and-modulus-for-computer-scientists/)
- [Wikipedia — Modular exponentiation (4^13 mod 497 example)](https://en.wikipedia.org/wiki/Modular_exponentiation)
- [Microsoft Excel MOD function](https://support.microsoft.com/office/mod-function-9b6cd169-b6ee-406a-a97b-edf2a9dc24f3)