#### RESULTS

Looking to execute modulo operations on numbers? Well, your search ends here! With this Modulo calculator, you can find the results of mod operations. All you have got to do is enter the initial number x and integer y in our calculator to determine the modulo number r.

## What exactly is Modulo?

Modulo or simply, mod, is a mathematical operation that determines the remainder when one integer is divided by another.

For integers **a** and **b**:

`\(\mathbf{a mod b = r}\)`

Where

**‘a’**stands for the dividend**‘b’**for divisor (or modulus)**‘r’**for the remainder

## A deeper look

To try to give you intuitive insight into the modulo operation without making stuff too complicated, we have illustrated the ‘clock’ example. Observe what happens when we increase numbers by one every time in this example.

The remainders begin from 0 and increase by 1 every time, until the point that the number reaches one less digit than the integer we are dividing by. When we reach this point, the sequence recycles.

With this illustration, we can picture the modulo operator in circles. We write 0 at the top of a circle and continuing clockwise writing integers 1, 2, 3, 4 ... up to one less than our modulus: in this case, 12 i.e, a clock with the 12 replaced with 0 would be the circle for a modulus of 12.

To find the result of ‘a mod b’, check out the steps given below:

Work out this clock for size B

Begin at 0 and wind the clock ‘A’ steps

Wherever we land is our solution.

(If our number is positive, we step clockwise, if it's negative, we step anti-clockwise.)

## How to calculate modulo?

Calculating mod becomes relatively easier once you’ve known the formula and understood the concept. All that’s left to do is find the remainder.

### mod formula

For integers a and b:

`\(\Large{\mathrm{a mod b = r}}\)`

Let’s do some practical work now and show you how our calculator determines the results of modulo operations.

Below are some of the examples where we calculate modulo.

## Modulo Calculation Example

Consider the following:

`\(\mathbf{Dividend} = a = 12 \)`

`\(\mathbf{Divisor} = b = 5\)`

`\(\mathbf{Remainder} = r =?\)`

we work our way up by subtracting the dividend from the product of the quotient and the divisor.

a mod b `\(= a-\Big(\Big\lbrack\dfrac{a}{b}\Big\rbrack\times b\Big)\)`

putting values in formula

`\(12 \mathrm{mod} 5 = 12-(\Big\lbrack\dfrac{12}{5}\Big\rbrack\times 5)\)`

When `\(12\)` divided by `\(5\)` gives us `\(2.4\)`, we round it down to the nearer number which in this case, is \(2\).

`\(12 \mathrm{mod} 5 = 12-(2\times5)\)`

Now multiplying the quotient and the divisor,

`\(12 \mathrm{mod} 5 = 12-10\)`

`\(\bold{12 \mathrm{mod} 5 = 52} \)`

Our modulo calculator can be used to determine the outcome between integer numbers of modulo operations.

The modulo operation, which is also frequently referred to as the modulus operation, identifies the remainder after dividing a given number by another number.

Apart from the above-mentioned expression, it can also be expressed as ‘a percent b’ in specific cases. On conventional calculators, you can determine modulo b using mod () function: mod (a, b) = r.

## Uses of modulo

### 1: Cryptography

Modular arithmetic is one of the most significant aspects of cryptography, which codes information through modulo operations that have an exponentially large modulus.

It supports public key systems directly in cryptography such as RSA and Diffie–Hellman and offers finite fields that support elliptical curves. This modular math is used in a variety of key symmetric algorithms including the International Data Encryption Algorithm (IDEA), RC4, and Advanced Encryption Standard (AES). RSA and Diffie–Hellman uses modular exponentiation.

### 2: Programming

Are you a geek who is into programming? Well, you might find it interesting that taking the modulo is how you can adjust items in a hash table:

- Say your table has N entries.
- Alter the item key to a number.
- Do mod N, and put the item in that bucket while keeping a linked list there.

As your hash table increases in size, you can re-compute the modulus for the keys.

### 3: Computational Applications

Modular arithmetic is generally used in computer algebra to limit the size of integer coefficients in intermediate calculations and data.

It is also employed to factorize polynomials, a problem for which all known optimal algorithms use modular arithmetic.

On the other end, modular arithmetic has profound usability for bitwise operations and others involving cyclic fixed-width data structures.

## Fun Fact

Ever heard of the ‘NP’ set of computational problems? It stands for the Non-deterministic Polynomial-time. NP is part of the problem space (P-SPACE) which contains problems related to computational complexity that are not solvable with deterministic

computers that take deterministic time for problem-solving. These are all the computers we know that can’t truly generate a random number due to the inherent determinism that governs our known laws of physics.

And it turns out that solving a system of non-linear arithmetic equations is an NP-complete problem which means that they would take forever to solve in deterministic time with deterministic computers. This is part of the elegance of modular arithmetic.