What is the Remainder of 4567 Divided by 8: A Detailed Explanation

Understanding the Problem and Solution

Today, we will explore the problem of determining the remainder when 4567 is divided by 8. This is a classic example that involves concepts of modular arithmetic and modular exponentiation. Let's start by breaking down the numbers involved and the process of finding the solution.

Factorization and Modular Properties

The first step in solving this problem is to understand the factorization of the base number:

456 23 × 3 × 19

Using this factorization, we can write:

4567 221 × 37 × 197

Since we are interested in the remainder when this number is divided by 8, we can simplify our task. Notice that:

23 is a factor of 456

This implies:

4567 ≡ 0 (mod 8)

Let's break this down further:

456 can be expressed as:

456 ≡ 0 (mod 8)

4567 can be simplified using modular exponentiation:

4567 ≡ 07 ≡ 0 (mod 8)

This confirms that 4567 divided by 8 leaves a remainder of 0.

Step-by-Step Demonstration

Let's go through the process step-by-step:

Express 456 in terms of its prime factors:

456 23 × 3 × 19

Then calculate the power:

4567 221 × 37 × 197

Now, let's simplify using the properties of modular arithmetic:

4567 (23)7 × 37 × 197

Since 23 is a factor of 8, and any power of 8 is still a multiple of 8:

221 23 × 218

This means:

221 ≡ 0 (mod 8)

Therefore:

4567 ≡ (23)7 × 37 × 197 ≡ 0 × 37 × 197 ≡ 0 (mod 8)

Thus, the remainder when 4567 is divided by 8 is:

0

Modular Arithmetic and Modular Exponentiation

Modular arithmetic is a fundamental concept in number theory and has numerous applications in modern cryptography, computer science, and many other fields. Modular exponentiation is a special case of modular arithmetic where an exponentiation operation is performed and the result is taken modulo some integer.

As an example, consider another modular exponentiation problem:

8 × 574566 574567

Here, we can see that:

57456 ≡ 0 (mod 8)

Thus:

574566 ≡ 0 (mod 8)

Which means:

574567 ≡ 0 (mod 8)

Hence, the remainder when 574567 is divided by 8 is also 0.

Conclusion

In conclusion, the remainder when 4567 is divided by 8 is 0. This is a result of the modular properties and the factorization of the base number. Understanding these concepts can help in solving similar problems in modular arithmetic and modular exponentiation.