# Modular Arithmetic

• Difficulty Level : Easy
• Last Updated : 04 May, 2020

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

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

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