Update Number Theory

docs/Number theory/Divisibility and Prime Numbers.md

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+---
+id: Divisibility-and-Prime-Numbers
+title: "Divisibility and Prime Numbers"
+sidebar_label: "Divisibility and Prime Numbers"
+sidebar_position: 10
+description: "A detailed guide to understanding and implementing the Divisibility and Prime Numbers."
+tags: [Divisibility, Prime Numbers, number theory, competitive programming]
+---
+
+# Divisibility and Prime Numbers
+
+Understanding divisibility and prime numbers is essential in number theory, with applications in cryptography, algorithm optimization, and more.
+
+## Divisibility
+
+Divisibility rules help quickly determine whether a number is divisible by another without performing full division. Here are some basic divisibility rules:
+
+1. **Divisibility by 2**: A number is divisible by 2 if its last digit is even.
+2. **Divisibility by 3**: A number is divisible by 3 if the sum of its digits is divisible by 3.
+    ```python
+    def is_divisible_by_3(n):
+        return sum(map(int, str(n))) % 3 == 0
+    ```
+3. **Divisibility by 5**: A number is divisible by 5 if its last digit is 0 or 5.
+    ```python
+    def is_divisible_by_5(n):
+    return str(n)[-1] in ('0', '5')
+    ```
+4. **Divisibility by 9**: A number is divisible by 9 if the sum of its digits is divisible by 9.
+    ```python
+    def is_divisible_by_9(n):
+    return sum(map(int, str(n))) % 9 == 0
+    ```
+5. **Divisibility by 10**: A number is divisible by 10 if it ends in 0.
+
+### Examples
+
+- **Is 123 divisible by 3?** Sum of digits: 1 + 2 + 3 = 6, which is divisible by 3, so 123 is also divisible by 3.
+- **Is 145 divisible by 5?** Last digit is 5, so it is divisible by 5.
+
+## Prime Numbers
+
+A **prime number** is a natural number greater than 1 that has no divisors other than 1 and itself.
+
+### Properties of Prime Numbers
+
+1. **Uniqueness**: Each natural number greater than 1 is either a prime itself or can be factored uniquely into prime numbers (Fundamental Theorem of Arithmetic).
+2. **Infinitude**: There are infinitely many prime numbers.
+
+### Prime Checking Algorithms
+
+1. **Trial Division**: Check if the number is divisible by any number up to its square root.
+2. **Sieve of Eratosthenes**: An efficient way to find all primes up to a certain limit.
+
+### Prime-Related Concepts
+
+- **Co-primes**: Two numbers are co-prime if their greatest common divisor (GCD) is 1.
+- **Prime Factorization**: Breaking down a number into the product of prime numbers. For example, 60 = 2 * 2 * 3 * 5.
+
+### Prime Number Theorems
+
+1. **Prime Density**: The density of prime numbers decreases as numbers get larger.
+2. **Approximation**: The Prime Number Theorem approximates the number of primes less than a given number \( n \) as \( n / \ln(n) \).
+
+## Practice Problems
+
+- [Check If Number Is Prime](https://leetcode.com/problems/check-if-number-is-prime/)
+- [Prime Factorization](https://www.geeksforgeeks.org/prime-factorization/)
+- [Sieve of Eratosthenes Implementation](https://leetcode.com/problems/count-primes/)
+- [Find GCD of Array (Co-primes)](https://leetcode.com/problems/find-greatest-common-divisor-of-array/)
+
+---
+
+By understanding divisibility rules and the properties of prime numbers, you can solve complex problems related to number theory more efficiently.
+
+

docs/Number theory/Modular Arithmetic.md

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+---
+id: Modular Arithmetic
+title: "Modular Arithmetic"
+sidebar_label: "Number theory"
+sidebar_position: 10
+description: "A detailed guide to understanding and implementing the Modular Arithmetic in Number Theory."
+tags: [Modular Arithmetic, number theory, competitive programming]
+---
+
+# Modular Arithmetic 
+
+Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after reaching a certain value—the modulus. It is commonly used in computer science and number theory. Below, we will cover the basics, properties, applications, and examples of modular arithmetic.
+
+## Basics of Modular Arithmetic
+
+In modular arithmetic, we work with the remainder when one integer is divided by another. We write:
+
+```
+a ≡ b (mod m)
+```
+to mean that **a** and **b** leave the same remainder when divided by **m**. This is equivalent to saying that **(a - b)** is divisible by **m**.
+
+### Example
+
+For example, if **a = 17**, **b = 5**, and **m = 12**, we find:
+
+```
+17 ≡ 5 (mod 12)
+```
+since 17 and 5 both leave a remainder of 5 when divided by 12.
+
+## Properties of Modular Arithmetic
+
+Modular arithmetic has several useful properties:
+
+1. **Addition**:
+   - `(a + b) % m = ((a % m) + (b % m)) % m`
+
+2. **Subtraction**:
+   - `(a - b) % m = ((a % m) - (b % m) + m) % m`
+
+3. **Multiplication**:
+   - `(a * b) % m = ((a % m) * (b % m)) % m`
+
+4. **Exponentiation**:
+   - `(a^b) % m = ((a % m)^b) % m`
+
+5. **Division (Inverse Modulo)**:
+   - Division is not directly supported, but you can divide by multiplying with the modular inverse of the divisor.
+
+## Applications of Modular Arithmetic
+
+Modular arithmetic is widely used in many fields, including:
+
+1. **Cryptography**: Algorithms like RSA use modular arithmetic to create secure encryption schemes.
+
+2. **Hashing Algorithms**: Modular arithmetic helps ensure that hash values are within a specific range.
+
+3. **Clock Arithmetic**: Modular arithmetic is used in clocks, as hours cycle back to 0 after reaching 12 or 24.
+
+4. **Random Number Generation**: It is also used to keep numbers within a certain range, which is crucial for generating random numbers.
+
+5. **Computer Graphics**: Modular arithmetic can help with cyclic transformations in graphics programming.
+
+## Modular Exponentiation
+
+Modular exponentiation is the process of finding `(base^exponent) % modulus` efficiently. This is especially useful in cryptography.
+
+### Fast Exponentiation Algorithm
+
+1. **Recursive Method**:
+   ```python
+   def mod_exp(base, exponent, modulus):
+       if exponent == 0:
+           return 1
+       half_exp = mod_exp(base, exponent // 2, modulus)
+       half_exp = (half_exp * half_exp) % modulus
+       if exponent % 2 != 0:
+           half_exp = (half_exp * base) % modulus
+       return half_exp
+
+    ```
+
+2. **Recursive Method**:
+   ```python
+   def mod_exp(base, exponent, modulus):
+    result = 1
+    base = base % modulus
+    while exponent > 0:
+        if (exponent % 2) == 1:
+            result = (result * base) % modulus
+        exponent = exponent >> 1
+        base = (base * base) % modulus
+    return result
+    ```
+## Modular Inverse
+To divide by a number under a modulus, we use the modular inverse. The modular inverse of a modulo m is a number x such that:
+```
+a * x ≡ 1 (mod m)
+```
+The modular inverse exists if and only if a and m are coprime.
+
+## Finding Modular Inverse
+1. **Using Extended Euclidean Algorithm:**
+
+```python
+    def extended_gcd(a, b):
+    if a == 0:
+        return b, 0, 1
+    gcd, x1, y1 = extended_gcd(b % a, a)
+    x = y1 - (b // a) * x1
+    y = x1
+    return gcd, x, y
+
+def mod_inverse(a, m):
+    gcd, x, _ = extended_gcd(a, m)
+    if gcd != 1:
+        return None  # Inverse does not exist
+    return x % m
+```
+2. **Using Fermat’s Little Theorem (when m is prime):**
+
+ If `m` is a prime number, the modular inverse of `a` can be calculated as:
+
+```
+a^(m-2) % m
+```
+## Practice Problems
+Here are some practice problems to solidify your understanding of modular arithmetic:
+
+- [Chinese Remainder Theorem](https://leetcode.com/problems/chinese-remainder-theorem/)
+- [Modulo Exponentiation](https://leetcode.com/problems/powx-n/)  
+- [Modular Multiplicative Inverse](https://www.geeksforgeeks.org/multiplicative-inverse-under-modulo-m/) 
+
+### Further Practice Resources
+
+For more extensive problem sets and further explanations, consider exploring:
+
+- [Modular Arithmetic Exercises on HackerRank](https://www.hackerrank.com/domains/tutorials/10-days-of-math)
+- [Advanced Modular Arithmetic Problems on CodeSignal](https://codesignal.com/interview-practice/)
+
+By solving these problems, you can deepen your understanding and mastery of modular arithmetic in various applications. 
+
+## Key Points to Remember
+- **Wrapping Around**: Numbers reset to 0 after reaching the modulus.
+- **Commutative and Associative Properties**: Useful for rearranging calculations in modular arithmetic.
+- **Negative Results**: For a result `(a - b) % m`, add `m` if the result is negative to keep within the correct range. 
+
+---
+