# Modular Arithmetic

**Modular arithmetic** is a notation and set of mathematics that were first introduced by Carl Friedrich Gauss. 🙂

The major insight is that equations can fruitfully be analyzed from the perspective of remainders. Standard equations use the ‘=’ sign. Modular arithmetic uses the ‘**≡**‘ sign. Two values that are ‘≡’ to each other are said to be congruent relative the modulus. In the case below, the modulus is 3. 🙂

Here’s an example of a modular equation:

7 ≡ 1 (mod 3).

By definition, this means that 3 divides 7 – 1.

**Definition 1**: a ≡ b (mod c) if and only if c divides a – b.

This definition tells us the following is true:

7 ≡ 1 ≡ 10 ≡ -2 (mod 3).

Now, one of the most interesting things about ‘≡’ is that it follows many of the same relations as ‘=’ . 🙂 🙂

**Notice 1**: For any value a,b,c,d,n where** a ≡ b (mod n) and c ≡ d (mod n): 🙂**

(a) **a + c ≡ b + d (mod n)**

** Proof :** We know that n divides (a + c) – (b + d) since this is equal to: (a -b) + (c – d).

(b) **a – c ≡ b – d (mod n)**

**Proof :** We know that n divides (a – c) – (b – d) since this is equal to: (a – b) – (c – d).

(c) **ac ≡ bd (mod n)**

**Proof :** We know that n divides ac – bd since this is equal to : c(a – b) + b(c – d).

(Q.E.D)

**Notice 2:** If **a ≡ b (mod n)** then:

(a)** a + c ≡ b + c (mod n)**

**Proof** : We know (a) since n divides a + c – (b + c) = a – b.

(b)** a – c ≡ b – c (mod n)**

** Proof** : We know (b) since n divides a – c – (b – c) = a – b.

(c) **ac ≡ bc (mod n)**

**Proof**: We know (c) since n divides ac – bc = c(a – b)

(Q.E.D)

**Corrolary 2.1**: **a ≡ d (mod n), b ≡ e (mod n), c ≡ f (mod n)**, then:

**a + b + c ≡ d + e + f mod n **

(1) We know that a + b ≡ d + e from above.

(2) We therefore know that (a + b) + c ≡ (d + e) + f.

(Q.E.D)

**Notice 3**: **a + b + c ≡ 0, a ≡ 0 (mod p), then b + c ≡ 0 (mod p).**

(1) a + b + c ≡ 0 (mod p) [Definition of ≡ ]

(2) b ≡ c (mod p) → a + b ≡ a + c (mod p) [See above]

(3) So, 0 ≡ a + b + c ≡ 0 + b + c ≡ b + c (mod p).

(Q.E.D)

I will cover modular arithmetic in depth 🙂 🙂

Today you must we wondering about its application ..

But after whole course You will surely love this topic

Thank you 🙂 🙂

If any doubts , then dont hesitate in asking Your doubts 🙂 🙂

I will surely love to solve your doubts

Posted on Tuesday,May 29, 2012, in Modular Arithmetic and tagged carl friedrich gauss, corrolary, lemma, mod, modular arithmetic, modulus, number theory, Prime, remainder, representation. Bookmark the permalink. 6 Comments.

Thank you for sharing your insights.

Welcome !!

Very instructive and good bodily structure of subject matter, now that’s user pleasant (:. 836889

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