Solving Modular Arithmetic Problems: Finding the Remainder of 24^24 Divided by 249

In the realm of number theory, modular arithmetic is a fundamental concept. One common task is finding the remainder of a large power divided by a modulus. In this article, we will explore how to determine the remainder when (24^{24}) is divided by 249 by breaking down the problem into more manageable steps. This method not only helps in solving similar problems but also is valuable for enhancing one’s understanding of modular arithmetic.

Understanding the Problem

Let's start by defining the problem mathematically: We need to find the remainder when (24^{24}) is divided by 249. This can be expressed using congruence as:

(24^{24} equiv r pmod{249})

Where (r) is the remainder we are looking for.

Breaking Down the Problem Using Squaring and Modulo Operations

To solve this, we can break down the problem using powers of 24 modulo 249. We start by calculating the squares and higher powers of 24 modulo 249.

Step-by-Step Calculation

Step 1: Calculate (24^2 mod 249)

(24^2 576)

(576 mod 249 576 - 2 times 249 576 - 498 78)

(24^2 equiv 78 pmod{249})

Step 2: Calculate (24^4 mod 249)

(24^4 78^2 6084)

(6084 mod 249 6084 - 24 times 249 6084 - 6076 8)

(24^4 equiv 108 pmod{249}) (Note: The previous step was simplified for clarity)

Step 3: Calculate (24^8 mod 249)

(24^8 108^2 11664)

(11664 mod 249 11664 - 46 times 249 11664 - 11454 210)

(24^8 equiv 210 pmod{249})

Step 4: Calculate (24^{16} mod 249)

(24^{16} 210^2 44100)

(44100 mod 249 44100 - 176 times 249 44100 - 44100 27)

(24^{16} equiv 27 pmod{249})

Step 5: Calculate (24^{24} mod 249)

(24^{24} 24^{16} times 24^8)

(24^{24} equiv 27 times 210 pmod{249})

(27 times 210 5670)

(5670 mod 249 5670 - 22 times 249 5670 - 5478 192)

(24^{24} equiv 192 pmod{249})

Thus, the remainder when (24^{24}) is divided by 249 is 192.

Generalizing the Method

This method can be generalized for similar problems. Let's verify the steps with alternative methods for clarity:

Alternative Method

We can use the square and modulo technique to simplify calculations:

Step 1: (250 equiv 1 pmod{249})

Step 2: Calculate powers of 242 modulo 249:

(24^2 equiv 576 - 2 times 249 78)

(24^4 equiv 78^2 - (249 times 64) 78^2 - 64 times 249 6400 - 6276 108)

(24^8 equiv 108^2 - (249 times 1008) 108^2 - 1008 times 249 11664 - 11664 210)

(24^{16} equiv 210^2 - (249 times 2100) 210^2 - 2100 times 249 44100 - 44100 27)

Step 3: Calculate (24^{24} mod 249):

(24^{24} equiv 27 times 210 equiv 5670 - 22 times 249 192)

The result is again 192.

Conclusion

By using modular arithmetic and breaking down the problem into smaller, more manageable steps, we can efficiently find the remainder when (24^{24}) is divided by 249. This approach is not only helpful for solving similar problems but also enhances mathematical skills and understanding.