Modular Arithmetic and Remainders: A Detailed Explanation of (3^{84} cdot 3^{63} cdot 3^{42} cdot 3^{21} cdot 1 mod 3^{20})

Understanding the concept of remainders and modular arithmetic can seem mystical, but it is a powerful tool in number theory, often found in advanced mathematics and cryptography. Let's explore a specific problem that involves these concepts: determining the remainder when (3^{84} cdot 3^{63} cdot 3^{42} cdot 3^{21} cdot 1) is divided by (3^{20}).

Step by Step Explanation of the Modular Arithmetic Solution

First, we define (x 3^{21}). This simplifies the original expression, making it easier to handle through modular arithmetic.

1. Simplifying the Expression

Given the expression (S 3^{84} cdot 3^{63} cdot 3^{42} cdot 3^{21} cdot 1), we can rewrite it using (x):

[S x^4 cdot x^3 cdot x^2 cdot x cdot 1]

2. Applying Modular Properties

Next, we need to simplify (S) modulo (3^{20}). Let's start by noting the property:

[3^{20} equiv -1 pmod{3^{20} cdot 1}]

This implies:

[3^{21} equiv -3 pmod{3^{20} cdot 1}]

3. Calculating Individual Terms

Using the equivalence (3^{21} equiv -3 pmod{3^{20} cdot 1}), we can calculate each power of (x):

Calculated Values:

(x^4 3^{21^4} equiv (-3)^4 81 pmod{3^{20} cdot 1}) (x^3 3^{21^3} equiv (-3)^3 -27 pmod{3^{20} cdot 1}) (x^2 3^{21^2} equiv (-3)^2 9 pmod{3^{20} cdot 1}) (x equiv -3 pmod{3^{20} cdot 1}) The constant term (1 pmod{3^{20} cdot 1}) is simply (1)

4. Substituting and Simplifying

Now, we substitute these values back into the expression for (S):

[S equiv 81 - 27 cdot 9 - 3 cdot 1 pmod{3^{20} cdot 1}]

Simplifying step by step:

First, (81 - 27 54) Then, (54 cdot 9 486 equiv 63 pmod{3^{20} cdot 1}) Next, (63 - 3 60) Finally, (60 cdot 1 60)

The constant term is simply (1), so:

[S equiv 60 1 equiv 61 pmod{3^{20} cdot 1}]

Conclusion

The remainder when (3^{84} cdot 3^{63} cdot 3^{42} cdot 3^{21} cdot 1) is divided by (3^{20}) is (61).

Summary

The process involves several key steps:

Simplifying the problem using modular properties. Finding individual powers of the base modulo the modulus. Substituting and simplifying the expression. Calculating the final remainder.

Understanding these steps not only helps solve this specific problem but also provides a foundation for more complex problems in modular arithmetic and number theory.