Important Applications on Linear Congruence

Number Theory

1. Divisibility
- 1.1. Greatest Common Divisor
  - 1.1.1. Important Problems On Greatest Common Divisor
- 1.2. Expressing Cubes As Differences Of Squares
2. Prime Numbers
- 2.1. Important Problems On Prime Numbers
- 2.2. Prime Number Theorem
  - 2.2.1. The Prime Number Theorem Explained With Interpretation
3. Congruence
- 3.1. Linear Congruence
  - 3.1.1. Important Applications On Linear Congruence
4. Number Theoretic Functions
- 4.1. Greatest Integer Functions
  - 4.1.1. Divisibility Of Binomial Coefficients
5. Quadratic Reciprocity
- 5.1. Legendre Symbol
  - 5.1.1. Legendre Symbol And Its Properties
6. Fermats Last Theorem
- 6.1. Proof Of Fermats Last Theorem
- 6.2. Pythagorean Triples
  - 6.2.1. Important Problems On Pythagorean Triples
7. Integers As Sum Of Squares
- 7.1. Sum Of More Than Two Squares
  - 7.1.1. Representation Of Integers As Sums Of Five Nonzero Squares For All Values Greater Than Or Equal To 170
- 7.2. Sum Of Two Squares
  - 7.2.1. Perfect Squares In Pythagorean Triples
8. Encryption And Decryption
- 8.1. Linear Cipher
  - 8.1.1. Important Applications On Linear Cipher
- 8.2. Hill Cipher
  - 8.2.1. Hill Cipher Decryption In Number Theory Step By Step With Example

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Important Applications

Linear Congruence is a key concept in Number Theory with wide-ranging applications in both theoretical and applied mathematics. This section focuses on the most important applications of Linear Congruence, including solving modular equations, cryptographic algorithms, and number-based problem-solving techniques. With step-by-step examples and real-world scenarios, this guide helps students understand how Linear Congruence is used in practice. Perfect for competitive exam preparation and academic learning, these applications enhance conceptual clarity and problem-solving skills.

Solve the linear congruence \( 3x \equiv 60 \pmod{98} \) using the Chinese Remainder Theorem without using modular inverse.

Answer:

We are given:

3x \equiv 60 \pmod{98}

Since \( 98 = 2 \cdot 49 \), we solve the congruence modulo 2 and modulo 49 separately.

Step 1: Solve modulo 2

\begin{align*} &3x \equiv 60 \pmod{2} \\ \Rightarrow \quad & x \equiv 0 \pmod{2} \\ \Rightarrow \quad & x = 2k,\quad k \in \mathbb{Z} \end{align*}

Step 2: Substitute into modulo 49

\begin{align*} & 3x \equiv 60 \pmod{49} \\ \Rightarrow \quad & 3(2k) \equiv 60 \pmod{49} \\ \Rightarrow \quad & 6k \equiv 11 \pmod{49} \quad (\text{since } 60 \equiv 11 \pmod{49}) \end{align*}

Step 3: Solve the linear congruence \( 6k \equiv 11 \pmod{49} \)

We convert this into a linear Diophantine equation:

\[ 6k - 49m = 11 \]

Use the Extended Euclidean Algorithm:

\begin{align*} & 49 = 8 \cdot 6 + 1 \\ \Rightarrow \quad & 1 = 49 - 8 \cdot 6 \\ \Rightarrow \quad & 1 = (-8) \cdot 6 + 1 \cdot 49 \\ \Rightarrow \quad & 11 = (-8 \cdot 11) \cdot 6 + (11 \cdot 49) \\ \Rightarrow \quad & 11 = (-88) \cdot 6 + 11 \cdot 49 \\ \Rightarrow \quad & 11 \equiv (-88) \cdot 6 \pmod{49} \\ \Rightarrow \quad & 6k \equiv (-88) \cdot 6 \pmod{49} \quad \text{since} \quad 6k \equiv 11 \pmod{49} \\ \Rightarrow \quad & k \equiv (-88) \pmod{49} \quad \text{since} \quad \gcd(6,49) = 1 \\ \Rightarrow \quad & k \equiv 10 \pmod{49} \quad \text{since} \quad -88 \equiv 10 \pmod{49} \end{align*}

Therefore:

\begin{align*} k &= 10 + 49t \\ x &= 2k = 2(10 + 49t) = 20 + 98t \end{align*}

Hence:

\boxed{x \equiv 20 \pmod{98}}