Introduction

In this article, we delve into the fascinating problem of determining the unit digit of a large power, specifically examining the case of (2137^{754}). This process involves understanding the unit digit of the base number and its periodicity in powers. We explore various methods to solve this problem, including mathematical properties and the concept of periodicity, to provide a comprehensive understanding of the solution.

Understanding Unit Digits and Modulus

The unit digit of a number is simply the digit at the one's place when the number is written in decimal form. For instance, the unit digit of 2137 is 7. This article will use the modulus operation, specifically modulo 10, to determine the unit digit of large powers. The modulus operation helps simplify the problem by reducing large numbers into a more manageable form.

The Problem: 2137754

We need to find the unit digit of the product (2137^{754}). Since the unit digit of 2137 is 7, our problem reduces to finding the unit digit of (7^{754}).

Identifying the Pattern in Powers of 7

The unit digits of the powers of 7 follow a repeating pattern. Let's explore this pattern in detail:

71 7 (unit digit is 7) 72 49 (unit digit is 9) 73 343 (unit digit is 3) 74 2401 (unit digit is 1) 75 16807 (unit digit is 7) 76 117649 (unit digit is 9) 77 823543 (unit digit is 3) 78 5764801 (unit digit is 1)

From this, we can see that the unit digits repeat every 4 powers: 7, 9, 3, 1. This periodicity can be used to determine the unit digit of any power of 7.

Solving (7^{754}) using Modulus and Periodicity

To find the unit digit of (7^{754}), we can use the fact that the unit digits repeat every 4 powers. We need to determine the position of 754 in this cycle:

[754 mod 4 2]

This means (7^{754}) has the same unit digit as (7^2). From our periodicity, the unit digit of (7^2) is 9. Therefore, the unit digit of (7^{754}) is 9.

Alternative Methods to Solve the Problem

There are several ways to solve this problem, each offering a different perspective. Here are a few methods:

Using the properties of exponents: Since (7^4 equiv 1 mod 10), we can express 754 as (4 times 188 2). Therefore,(7^{754} equiv 7^{4 times 188 2} equiv (7^4)^{188} times 7^2 equiv 1^{188} times 7^2 equiv 7^2 equiv 9 mod 10).

Identifying the cyclic nature directly: The unit digit of any power of 7 repeats every 4 numbers following the sequence 7, 9, 3, 1. Since 754 is 2 more than a multiple of 4, the unit digit of (7^{754}) is the same as the unit digit of (7^2), which is 9.

Simplifying the process: The unit digit of (2137) is 7. The unit digits of the powers of 7 are 7, 9, 3, 1. Since 754 divided by 4 leaves a remainder of 2, the unit digit of (7^{754}) is the same as the unit digit of (7^2), which is 9.

Conclusion

In conclusion, the unit digit of the product (2137^{754}) is 9. This solution utilizes the periodicity of unit digits in powers and the properties of exponents to provide a step-by-step approach to solving this intriguing problem.