# Modular Arithmetic

Modular arithmetic is the branch of arithmetic mathematics related with the “mod” functionality. Basically, modular arithmetic is related with computation of “mod” of expressions. Expressions may have digits and computational symbols of addition, subtraction, multiplication, division or any other. Here we will discuss briefly about all modular arithmetic operations.

**Quotient Remainder Theorem :**

It states that, for any pair of integers a and b (b is positive), there exists two unique integers q and r such that:

a = b x q + r

where 0 <= r < b

**Example:**

If a = 20, b = 6

then q = 3, r = 2

20 = 6 x 3 + 2

**Modular Addition :**

Rule for modular addition is:

(a + b) mod m = ((a mod m) + (b mod m)) mod m

**Example:**

(15 + 17) % 7 = ((15 % 7) + (17 % 7)) % 7 = (1 + 3) % 7 = 4 % 7 = 4

Same rule is for modular subtraction. We don’t require much modular subtraction but it can also be done in same way.

**Modular Multiplication :**

Rule for modular multiplication is:

(a x b) mod m = ((a mod m) x (b mod m)) mod m

**Example:**

(12 x 13) % 5 = ((12 % 5) x (13 % 5)) % 5 = (2 x 3) % 5 = 6 % 5 = 1

**Modular Division :**

Modular division is totally different from modular addition, subtraction and multiplication. It also does not exist always.

(a / b) mod m is not equal to ((a mod m) / (b mod m)) mod m.

This is calculated using following formula:

(a / b) mod m = (a x (inverse of b if exists)) mod m

**Modular Inverse :**

The modular inverse of a mod m exists only if a and m are relatively prime i.e. gcd(a, m) = 1.

Hence, for finding inverse of a under modulo m,

if (a x b) mod m = 1 then b is modular inverse of a.

**Example:**

a = 5, m = 7

(5 x 3) % 7 = 1

hence, 3 is modulo inverse of 5 under 7.

**Modular Exponentiation :**

Finding a^b mod m is the modular exponentiation. There are two approaches for this – recursive and iterative.

**Example:**

a = 5, b = 2, m = 7 (5 ^ 2) % 7 = 25 % 7 = 4

Below are some more important concepts related to Modular Arithmetic

- Euler’s Totient Function
- Compute n! under modulo p
- Wilson’s Theorem
- How to compute mod of a big number?
- Find value of y mod (2 raised to power x)

**
Recent Articles on Modular Arithmetic**.

Don’t stop now and take your learning to the next level. Learn all the important concepts of Data Structures and Algorithms with the help of the most trusted course: DSA Self Paced. Become industry ready at a student-friendly price.

## Recommended Posts:

- Modular Exponentiation (Power in Modular Arithmetic)
- Modular Addition
- Modular Multiplication
- Modular Division
- Trick for modular division ( (x1 * x2 .... xn) / b ) mod (m)
- Modular exponentiation (Recursive)
- Modular multiplicative inverse
- Modular multiplicative inverse from 1 to n
- Number of solutions to Modular Equations
- Modular Exponentiation of Complex Numbers
- How to avoid overflow in modular multiplication?
- Find modular node in a linked list
- Using Chinese Remainder Theorem to Combine Modular equations
- Smallest number to be added in first Array modulo M to make frequencies of both Arrays equal

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.