# Category Archives: Modular Arithmetic

## 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