Understanding the Last Digit of Complex Exponents: A Deep Dive

When dealing with complex exponents, determining the last digit of a number can be both interesting and challenging. In this article, we will explore a specific example to understand how to find the last digit of 3^{5^{7^{9}} - 1} and discuss the underlying mathematical principles involved.

Introduction to Euler's Theorem and Modulo Arithmetic

To understand the problem, we need to familiarize ourselves with some fundamental concepts in number theory, specifically Euler's theorem and modulo arithmetic. Euler's theorem states that if two numbers a and n are coprime, then

a^{?(n)} ≡ 1 (mod n), where ?(n) is the Euler's totient function.

Step-by-Step Analysis

Step 1: Simplify the Expression

First, we simplify the exponent of the expression 3^{5^{7^{9}} - 1} by breaking it down into smaller, more manageable components.

Using the property that 3^5 ≡ 3 (mod 10), we can simplify further.

Step 2: Apply Euler's Theorem

We need to find the exponent modulo 4, as the totient of 10 is 4 (i.e., ?(10) 4). We use Euler's theorem to simplify the base raised to a large exponent:

5^{7^9} ≡ 1 (mod 4)

This implies that

3^{5^{7^9}} ≡ 3^1 ≡ 3 (mod 10)

Step 3: Compute the Final Expression

Combining the results, we now evaluate 3^{5^{7^9}} - 1:

3^{5^{7^9}} - 1 ≡ 3 - 1 ≡ 2 ≡ 8 (mod 10)

Hence, the last digit of 3^{5^{7^9} - 1} is 8.

Explanation and Further Exploration

The last digit of powers of three follows a cycle of four: 3, 9, 7, 1. This repeating pattern can help us determine the last digit of larger powers of three by dividing the exponent by 4 and using the remainder. For the base seven, the cycle is 7, 9, 3, 1. Understanding these patterns simplifies the problem significantly.

Example Walkthrough

1. Calculate 3^5:

3^5 243 ≡ 3 (mod 10)

2. Calculate 7^2 and simplify:

7^2 49 ≡ 49 - 50 1 -1 (mod 10)

3. Calculate 3^{5^7} - 1 using the simplified exponent:

(7^2) 49 ≡ -1 (mod 10)

4. Calculate the final expression:

3^{5^{7^{9}} - 1} ≡ 3^3 - 1 ≡ 7 - 1 ≡ 8 (mod 10)

Thus, the last digit of 3^{5^{7^{9}} - 1} is 8.

Conclusion

By understanding Euler's theorem and modulo arithmetic, we can efficiently determine the last digit of complex exponents. The key insight is recognizing the repeating patterns in the last digits of powers and applying theorems to simplify the problem.