Exploring Modular Arithmetic: A Deep Dive into 21000 * 32000 * 54000 Mod 13

Modular arithmetic is a core concept in number theory and cryptography, involving operations performed on integers where results are taken modulo a fixed number. In this article, we will delve into a complex modular arithmetic problem to understand the elegance and power of Fermat's Little Theorem and how it simplifies such problems.

Understanding the Problem

The problem at hand is to find the value of 21000 * 32000 * 54000 modulo 13. To solve this, we exploit the properties of modular arithmetic and Fermat's Little Theorem.

Fermat's Little Theorem and Its Applications

Fermat's Little Theorem states that if p is a prime number, then for any integer a, we have ap – 1 ≡ 1 (mod p). This theorem is particularly useful when the exponent is a multiple of p – 1.

Breaking Down the Exponents

First, let's express the exponents in terms of their remainders modulo 12:

1000 modulo 12 gives 4, so 21000 ≡ 24 (mod 13)

2000 modulo 12 gives 8, so 32000 ≡ 38 (mod 13)

4000 modulo 12 gives 4, so 54000 ≡ 54 (mod 13)

Evaluating the Exponents Step-by-Step

Let's evaluate each of these parts individually:

24 ≡ 16 (mod 13) and since 16 mod 13 ≡ 3, we have 24 ≡ 3 (mod 13)

38 ≡ 6561 (mod 13). Breaking this down further, 6561 mod 13 ≡ 9 (note: 6561 38)

54 ≡ 625 (mod 13). Breaking this down further, 625 mod 13 ≡ 1

Combining the Results

Now, combining these results, we have:

21000 * 32000 * 54000 ≡ 3 * 9 * 1 (mod 13)

3 * 9 * 1 ≡ 27 (mod 13) and 27 mod 13 ≡ 1 (because 27 - 2 * 13 1)

Subsequently, this simplifies to 13 mod 13 ≡ 0.

Conclusion

By applying Fermat's Little Theorem, we were able to greatly simplify the problem of finding 21000 * 32000 * 54000 mod 13. The solution reveals that the expression is congruent to 0 modulo 13. This showcases the power and utility of modular arithmetic in solving complex problems efficiently.