Modulo Calculator

The modulo operation answers the question of what is left over when you divide one number $a$ by another number $n$. The remainder is called the "modulo" and is represented as $a \bmod n$. For example: $17 \bmod 5 = 2$. If you split 17 into groups of five, you get three full groups with two remaining.

This calculator will give you the remainder, quotient and full breakdown of a = n × q + r when you input the dividend and divisor above. It supports negative numbers and decimals, and it shows if the result of the modulo operation is not the same as "remainder" used by programmers. Switch to congruence mode to check if two numbers have the same remainder.

What is a modulo operation?

Let's start with a normal division with remainder. When you divide $a$ (dividend) by $n$ (divisor, also called modulus), you get an integer quotient $q$ and a remainder $r$, such that $a = nq + r$. In modulo operation we keep only the remainder and discard the quotient.

In a strict definition, the rounding function is used, i.e., rounded to the nearest integer.

The formula for modulo operation is $a \bmod n = a - n\left\lfloor \dfrac{a}{n} \right\rfloor$.

If $n$ is a positive number, the result will always be within the range of $0 \le r < n$. So the modulo operation never exceeds the divisor. To easily check this, make sure that $a \bmod n$ is always less than $n$ itself.

When the divisor is larger than the dividend, it's a very simple special case. Since no quotient can be calculated, the remainder is equal to the dividend itself. This is because 2 and 10 both occur zero times in their respective dividends, so $1 \bmod 2 = 1$ and $5 \bmod 10 = 5$.

How to calculate modular arithmetic manually.

This method is based on long division which you probably already know how to do. You can turn a division problem into a modulo operation with four steps.

Let's take $23 \bmod 7$ as an example.

1. I divide: 23 ÷ 7 = 3.285...

2. I round down to get a quotient of 3.

3. I multiply back. So, 7 times 3 is equal to 21.

4. You subtract and determine the remainder. So it is $23 \bmod 7 = 2$.

The instructions on this page will perform these steps precisely for each value entered. The last line recalculates $a$ based on $n \times q + r$, and verifies the result.

Some example calculations:

Below are some of the common questions or issues that may be addressed by this calculator.

Modulo operations with negative numbers:

Negative numbers are an area where the modulo calculation tool shows its strengths because two apparently plausible conventions do not agree. The rounding definition given above still applies. For example, if we consider $-7 \bmod 3$, the quotient $\lfloor -7/3 \rfloor$ is rounded to $-3$, which gives a remainder of $-7 - 3(-3) = 2$. According to the mathematical definition, the result would be $-7 \bmod 3 = 2$, a positive number.

Most programming languages do not use this result. In C, Java and JavaScript the operator $\%$ truncates the quotient towards zero, so that $-7 \% 3 = -1$. Both methods are consistent within their respective rules but they answer slightly different questions. A useful rule of thumb is that modulo with rounding takes the sign of the divisor while truncated remainder takes the sign of the dividend.

This calculator primarily shows the result of the mathematical (rounded) modulo operation. If the results differ due to a minus sign, for each case the rounded remainder is shown next to it so that you can use the result accepted in your language.

Modular congruence:

Two numbers $a$ and $b$ are congruent modulo $n$ if they have the same remainder. This is denoted by $a \equiv b \pmod{n}$. Another equivalent condition is that the difference $a - b$ is a multiple of $n$.

For example: $17 \equiv 12 \pmod 5$. Both remainders are 2 and the difference $17 - 12 = 5$ is a multiple of 5. The same applies to: $9 \equiv 21 \pmod 6$. $21 - 9 = 12$ is a multiple of 6.

In the congruence mode this is checked and all numbers that have the same remainder are listed as a set called a congruence class. Any number obtained by adding or subtracting the modulus belongs to this set.

Rules for modulo arithmetic.

Congruence relations work well with addition, subtraction and multiplication. That's why modular arithmetic is so useful. For positive moduli we have:

$(a + b) \bmod n = \big((a \bmod n) + (b \bmod n)\big) \bmod n$

$(a \times b) \bmod n = \big((a \bmod n) \times (b \bmod n)\big) \bmod n$

This means that you can apply the modulo operation separately to each part and then combine them. This keeps the numbers small. This is the secret of how computers can compute extremely large numbers in cryptography without having to store them in their full form.

Examples of applications for modular arithmetic.

A very common example is the clock. The 12-hour format is based on modulo-12 arithmetic. Four hours after 11 o'clock it's $(11 + 4) \bmod 12 = 3$. Also, days of the week work in a similar way with the modulo-7 system.

Besides timing, the modulo operation is one of the most commonly used operations in programming. By calculating n mod 2, you can determine whether a number is even or odd. With i mod length, an index can be kept within the bounds of an array. Hash tables, round-robin schedulers, checksums for credit cards and ISBNs, and public-key cryptography all rely on this same operation of cyclical repetition.

About division by zero:

The divisor $n$ can be positive, negative or a decimal number but cannot be zero. As division by zero is undefined, so is $a \bmod 0$. If you enter zero as the divisor, this calculator will not give an instant result but instead explicitly inform you about it to avoid any confusion.