Basic Modular Arithmetic

In simple terms, Modular Arithmetic calculates the remainder of anything divided by anything. The later is called modulus. e.g. ( 15 / 7 ) Quotient: 2, Remainder 1. 

In greater sense, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus ( src: Wikipedia ).

Take a look at hour clock. After 12:59 pm, we say 1:00 pm, because we modulo the hours by 12. 13 pm % 12 is 1 pm. (%) represents modulus operator.
Examples: 5 % 2 = 1, 14458948 % 25 = 23

a≡b (mod n)

This says that a $A$is congruent to b modulo n. It means both a and b has same remainder when divided by n. e.g. 38≡14 (mod 12)

Commonly Used Properties:

• Reflexivity: a ≡ a (mod n)
• Symmetry: a ≡ b (mod n) if and only if b ≡ a (mod n)
• Transitivity: If a ≡ b (mod n) and b ≡ c (mod n), then a ≡ c (mod n)
• If a₁ ≡ b₁ (mod n) and a₂ ≡ b₂ (mod n), or if a ≡ b (mod n), then:a + k ≡ b + k (mod n) for any integer k (compatibility with translation)
• k a ≡ k b (mod n) for any integer k (compatibility with scaling)
• a₁ + a₂ ≡ b₁ + b₂ (mod n) (compatibility with addition)
• a₁ – a₂ ≡ b₁ – b₂ (mod n) (compatibility with subtraction)
• a₁ a₂ ≡ b₁ b₂ (mod n) (compatibility with multiplication)
• a^k ≡ b^k (mod n) for any non-negative integer k (compatibility with exponentiation)

Simple C program to perform Modulo:

	#include <iostream>
	using namespace std;
	int main() {
	int a = 5 % 2;
	printf("%d\n",a);
	return 0;
	}

view raw ModulerArithmatic.c hosted with ❤ by GitHub

In CS, modular arithmetic is used in many algorithms where integers become too big to handle.

For more details:

https://en.wikipedia.org/wiki/Modular_arithmetic

https://www.khanacademy.org/computing/computer-science/cryptography/modarithmetic/a/what-is-modular-arithmetic

$B$

 modulo $C$C.