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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 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)
  • ak ≡ bk (mod n) for any non-negative integer k (compatibility with exponentiation)
Simple C program to perform Modulo:

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

For more details:

 modulo C.

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