The Last Non-Zero Digit of 2^{5^7} × 3^4 × 5^{5^3}

Introduction

Understanding the last non-zero digit of a complex expression is an interesting problem in number theory and digital analysis. In this article, we explore the last non-zero digit of the expression 2^{5^7} × 3^4 × 5^{5^3}. This problem not only involves intricate exponentiation but also modulo arithmetic, providing a rich ground to delve into the intricacies of number theory.

Step-by-Step Solution of the Expression

The first step in solving 2^{5^7} × 3^4 × 5^{5^3} is to simplify the expression through breaking down the components. Let's start by analyzing the expression:

2^{5^7} × 3^4 × 5^{5^3} 2^{5^{3 4}} × 3^4 × 5^{5^3}

Breaking this down further:

2^{5^7} × 3^4 × 5^{5^3} 2^{5^4} × 2^{5^3} × 3^4 × 5^{5^3}

Further simplifying:

2^{5^4} × 3^4 × 5^{5^3} × 2^{5^3}

Given that 10^{5^3} does not affect the last non-zero digit:

2^{5^4} × 3^4 × 5 × 2^{5^3} 2^{5^4} × 3^4 × 10^{5^3}

The Role of 10^{5^3}

Since 10^{5^3} does not influence the last non-zero digit, we ignore this factor:

The last non-zero digit of the expression is derived from 2^{5^4} × 3^4.

Unit Digit Analysis

The unit digit of a product is the same as the unit digit of the product of the unit digits of the factors. We focus on the unit digits of 2^{5^4} and 3^4.

Modulo 10 Analysis

First, we calculate:

3^4 ≡ 81 ≡ 1 (mod 10), which means 3^4 ≡ 1 (mod 10) [Equation 1] 2^5 ≡ 2 (mod 10) (2^5)^5 ≡ 2^5 (mod 10) 2^{25} ≡ 2 (mod 10) (2^{25})^5 ≡ 2^5 (mod 10) 2^{125} ≡ 2 (mod 10) (2^{125})^5 ≡ 2^5 (mod 10) 2^{625} ≡ 2 (mod 10) [Equation 2]

By combining Equations 1 and 2:

2^{5^4} × 3^4 ≡ 2 × 1 ≡ 2 (mod 10)

This implies that the last non-zero digit of 2^{5^7} × 3^4 × 5^{5^3} is 2.

Alternative and Simplified Approach

In a more direct approach: 2^{5^7} × 5^{5^3} simplifies to:

2^{5^4-15^3} × 10^{5^3}

Which leaves us with a simpler expression:

81 × 2^{624 × 125}.

The term 81 does not affect the last digit, leaving us with:

2^{624 × 125} 2^{78000}

The last digits of the powers of 2 cycle through 2, 4, 8, 6. Since 78000 ≡ 0 (mod 4), the last digit of 2^{78000} is 6.

Therefore, the last non-zero digit of the expression 2^{5^7} × 3^4 × 5^{5^3} is 6.