Properties of Modular Arithmetic

Oct 13, 2025 by miodeya.com 97 views

Introduction

In this article, we will explore the concept of modular arithmetic and its application in determining the remaining number of a large number when divided by another number. We will specifically focus on finding the remainder of the sum of 2^2024 and 10^2024 when divided by 36.

Modular Arithmetic

Modular arithmetic is a system of arithmetic that "wraps around" after reaching a certain value, called the modulus. In other words, it is a way of performing arithmetic operations on integers, where the result of an operation is the remainder of the division of the result by the modulus.

Properties of Modular Arithmetic

Modular arithmetic has several properties that make it useful in various applications. Some of the key properties include:

Closure: The result of an operation in modular arithmetic is always an integer.
Associativity: The order in which operations are performed does not affect the result.
Commutativity: The order in which operands are used does not affect the result.
Distributivity: The distributive property holds in modular arithmetic.

Modular Exponentiation

Modular exponentiation is a technique used to compute the result of raising a number to a power modulo another number. It is a fundamental concept in number theory and has numerous applications in cryptography, coding theory, and other areas.

The Chinese Remainder Theorem

The Chinese Remainder Theorem (CRT) is a fundamental result in number theory that provides a way to solve systems of congruences. It states that if we have a system of congruences:

x ≡ a1 (mod n1) x ≡ a2 (mod n2) ... x ≡ ak (mod nk)

where ni are pairwise coprime, then there exists a unique solution modulo N = n1n2...nk.

Applying Modular Arithmetic to 2^2024 + 10^2024

To determine the remaining number of 2^2024 + 10^2024 by 36, we can use the properties of modular arithmetic. We can first find the remainder of 2^2024 and 10^2024 when divided by 36 separately, and then add the results to get the final remainder.

Finding the Remainder of 2^2024

To find the remainder of 2^2024 when divided by 36, we can use the property of modular exponentiation. We can write:

2^2024 ≡ (2⁴⁾502 * 2^4 (mod 36)

Since 2^4 ≡ 16 (mod 36), we have:

2^2024 ≡ 16^502 * 16 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

16^502 ≡ (16²⁾251 (mod 36) 16^2 ≡ 256 ≡ 4 (mod 36) 16^502 ≡ 4^251 (mod 36)

Since 4^2 ≡ 16 (mod 36), we have:

4^251 ≡ (4²⁾125 * 4 (mod 36) 4^251 ≡ 16^125 * 4 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

16^125 ≡ (16²⁾62 * 16 (mod 36) 16^125 ≡ 4^62 * 16 (mod 36)

Since 4^2 ≡ 16 (mod 36), we have:

4^62 ≡ (4²⁾31 (mod 36) 4^62 ≡ 16^31 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

16^31 ≡ (16²⁾15 * 16 (mod 36) 16^31 ≡ 4^15 * 16 (mod 36)

Since 4^2 ≡ 16 (mod 36), we have:

4^15 ≡ (4²⁾7 * 4 (mod 36) 4^15 ≡ 16^7 * 4 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

16^7 ≡ (16²⁾3 * 16 (mod 36) 16^7 ≡ 4^3 * 16 (mod 36)

Since 4^2 ≡ 16 (mod 36), we have:

4^3 ≡ 4^2 * 4 (mod 36) 4^3 ≡ 16 * 4 (mod 36) 4^3 ≡ 64 ≡ 28 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

16^7 ≡ 28 * 16 (mod 36) 16^7 ≡ 448 ≡ 8 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

4^15 ≡ 8 * 4 (mod 36) 4^15 ≡ 32 ≡ 28 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

16^31 ≡ 28 * 16 (mod 36) 16^31 ≡ 448 ≡ 8 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

4^62 ≡ 8 * 4 (mod 36) 4^62 ≡ 32 ≡ 28 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

16^125 ≡ 28 * 16 (mod 36) 16^125 ≡ 448 ≡ 8 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

4^251 ≡ 8 * 4 (mod 36) 4^251 ≡ 32 ≡ 28 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

16^502 ≡ 28 * 16 (mod 36) 16^502 ≡ 448 ≡ 8 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

2^2024 ≡ 8 * 16 (mod 36) 2^2024 ≡ 128 ≡ 8 (mod 36)

Finding the Remainder of 10^2024

To find the remainder of 10^2024 when divided by 36, we can use the property of modular arithmetic. We can write:

10^2024 ≡ (10⁴⁾502 * 10^4 (mod 36)

Since 10^4 ≡ 10000 ≡ 4 (mod 36), we have:

10^2024 ≡ 4^502 * 4 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

4^502 ≡ (4²⁾251 (mod 36) 4^502 ≡ 16^251 (mod 36)

Since 16^2 ≡ 256 ≡ 4 (mod 36), we have:

16^251 ≡ (16²⁾125 * 16 (mod 36) 16^251 ≡ 4^125 * 16 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

4^125 ≡ (4²⁾62 * 4 (mod 36) 4^125 ≡ 16^62 * 4 (mod 36)

Since 16^2 ≡ 256 ≡ 4 (mod 36), we have:

16^62 ≡ (16²⁾31 (mod 36) 16^62 ≡ 4^31 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

4^31 ≡ (4²⁾15 * 4 (mod 36) 4^31 ≡ 16^15 * 4 (mod 36)

Since 16^2 ≡ 256 ≡ 4 (mod 36), we have:

16^15 ≡ (16²⁾7 * 16 (mod 36) 16^15 ≡ 4^7 * 16 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

4^7 ≡ (4²⁾3 * 4 (mod 36) 4^7 ≡ 16^3 * 4 (mod 36)

Since 16^2 ≡ 256 ≡ 4 (mod 36), we have:

16^3 ≡ 16^2 * 16 (mod 36) 16^3 ≡ 4 * 16 (mod 36) 16^3 ≡ 64 ≡ 28 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

4^7 ≡ 28 * 4 (mod 36) 4^7 ≡ 112 ≡ 28 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

16^15 ≡ 28 * 16 (mod 36) 16^15 ≡ 448 ≡ 8 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

**Determine the Remaining Number of 2^2024 + 10^2024 by 36** ===========================================================

Q&A

Q: What is modular arithmetic?

A: Modular arithmetic is a system of arithmetic that "wraps around" after reaching a certain value, called the modulus. In other words, it is a way of performing arithmetic operations on integers, where the result of an operation is the remainder of the division of the result by the modulus.

Q: What are the properties of modular arithmetic?

A: Modular arithmetic has several properties that make it useful in various applications. Some of the key properties include:

Closure: The result of an operation in modular arithmetic is always an integer.
Associativity: The order in which operations are performed does not affect the result.
Commutativity: The order in which operands are used does not affect the result.
Distributivity: The distributive property holds in modular arithmetic.

Q: What is modular exponentiation?

A: Modular exponentiation is a technique used to compute the result of raising a number to a power modulo another number. It is a fundamental concept in number theory and has numerous applications in cryptography, coding theory, and other areas.

Q: How do you find the remainder of 2^2024 when divided by 36?

A: To find the remainder of 2^2024 when divided by 36, we can use the property of modular exponentiation. We can write:

2^2024 ≡ (2⁴⁾502 * 2^4 (mod 36)

Since 2^4 ≡ 16 (mod 36), we have:

2^2024 ≡ 16^502 * 16 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

16^502 ≡ (16²⁾251 (mod 36) 16^502 ≡ 256^251 (mod 36)

Since 256 ≡ 4 (mod 36), we have:

256^251 ≡ 4^251 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

4^251 ≡ (4²⁾125 * 4 (mod 36) 4^251 ≡ 16^125 * 4 (mod 36)

Since 16^2 ≡ 256 ≡ 4 (mod 36), we have:

16^125 ≡ (16²⁾62 * 16 (mod 36) 16^125 ≡ 4^62 * 16 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

4^62 ≡ (4²⁾31 (mod 36) 4^62 ≡ 16^31 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

16^31 ≡ (16²⁾15 * 16 (mod 36) 16^31 ≡ 4^15 * 16 (mod 36)

Since 4^2 ≡ 16 (mod 36), we have:

4^15 ≡ (4²⁾7 * 4 (mod 36) 4^15 ≡ 16^7 * 4 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

16^7 ≡ (16²⁾3 * 16 (mod 36) 16^7 ≡ 4^3 * 16 (mod 36)

Since 4^2 ≡ 16 (mod 36), we have:

4^3 ≡ 4^2 * 4 (mod 36) 4^3 ≡ 16 * 4 (mod 36) 4^3 ≡ 64 ≡ 28 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

16^7 ≡ 28 * 16 (mod 36) 16^7 ≡ 448 ≡ 8 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

4^15 ≡ 8 * 4 (mod 36) 4^15 ≡ 32 ≡ 28 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

16^31 ≡ 28 * 16 (mod 36) 16^31 ≡ 448 ≡ 8 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

4^62 ≡ 8 * 4 (mod 36) 4^62 ≡ 32 ≡ 28 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

16^125 ≡ 28 * 16 (mod 36) 16^125 ≡ 448 ≡ 8 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

4^251 ≡ 8 * 4 (mod 36) 4^251 ≡ 32 ≡ 28 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

16^502 ≡ 28 * 16 (mod 36) 16^502 ≡ 448 ≡ 8 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

2^2024 ≡ 8 * 16 (mod 36) 2^2024 ≡ 128 ≡ 8 (mod 36)

Q: How do you find the remainder of 10^2024 when divided by 36?

A: To find the remainder of 10^2024 when divided by 36, we can use the property of modular exponentiation. We can write:

10^2024 ≡ (10⁴⁾502 * 10^4 (mod 36)

Since 10^4 ≡ 10000 ≡ 4 (mod 36), we have:

10^2024 ≡ 4^502 * 4 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

4^502 ≡ (4²⁾251 (mod 36) 4^502 ≡ 16^251 (mod 36)

Since 16^2 ≡ 256 ≡ 4 (mod 36), we have:

16^251 ≡ (16²⁾125 * 16 (mod 36) 16^251 ≡ 4^125 * 16 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

4^125 ≡ (4²⁾62 * 4 (mod 36) 4^125 ≡ 16^62 * 4 (mod 36)

Since 16^2 ≡ 256 ≡ 4 (mod 36), we have:

16^62 ≡ (16²⁾31 (mod 36) 16^62 ≡ 4^31 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

4^31 ≡ (4²⁾15 * 4 (mod 36) 4^31 ≡ 16^15 * 4 (mod 36)

Since 16^2 ≡ 256 ≡ 4 (mod 36), we have:

16^15 ≡ (16²⁾7 * 16 (mod 36) 16^15 ≡ 4^7 * 16 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

4^7 ≡ (4²⁾3 * 4 (mod 36) 4^7 ≡ 16^3 * 4 (mod 36)

Since 16^2 ≡ 256 ≡ 4 (mod 36), we have:

16^3 ≡ 16^2 * 16 (mod 36) 16^3 ≡ 4 * 16 (mod 36) 16^3 ≡ 64 ≡ 28 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

4^7 ≡ 28 * 4 (mod 36) 4^7 ≡ 112 ≡ 28 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

16^15 ≡ 28 * 16 (mod 36) 16^15 ≡ 448 ≡ 8 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

4^31 ≡ 8 * 4 (mod 36) 4^31 ≡ 32 ≡ 28 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

16^62 ≡ 28 * 16 (mod 36) 16^62 ≡ 448 ≡ 8 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

4^125 ≡ 8 * 4 (mod 36) 4^125 ≡ 32 ≡ 28 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

16^251 ≡ 28 * 16 (mod 36) 16^251 ≡ 448 ≡ 8 (mod 36)

Now, we can use the property of modular arithmetic to simplify the expression:

4^502 ≡ 8 *