Determining the Last Digit of 112021 Modulo 10: A Comprehensive Guide

In this article, we will explore a detailed method to find the last digit of the expression 112021 modulo 10. This problem is a classic example in number theory and can be approached through several steps involving pattern recognition and modular arithmetic. Let's break down the solution to understand it thoroughly.

Understanding the Problem

The question asks for the last digit of 112021. To solve this, we can use the concept of modulo operation, specifically modulo 10, which helps in identifying the last digit of large numbers. We start by considering the base number 11 and its powers.

Observing Patterns in Powers of 11

Let's observe the last digits of the first few powers of 11:

111 11 (last digit is 1) 112 121 (last digit is 1) 113 1331 (last digit is 1) 114 14641 (last digit is 1)

From this, we can see a clear pattern: the last digit of 11n is 1 for any positive integer n. This pattern simplifies our problem significantly.

Mathematical Representation

To formalize this observation, we express it using congruence notation:

112021 ≡ x (mod 10)

From our pattern observation, we can deduce:

11 ≡ 1 (mod 10)

Using the property of congruence, we can elevate both sides to the power of 2021:

112021 ≡ 12021 (mod 10)

Since 1 raised to any power is still 1, we have:

112021 ≡ 1 (mod 10)

The Final Answer

Therefore, the last digit of 112021 is 1.

Conclusion

In conclusion, by understanding and applying the properties of modulo operations and recognizing patterns in powers of 11, we can easily determine the last digit of 112021 to be 1. This problem serves as a great example to showcase the power of number theory in solving seemingly complex mathematical problems.

Keywords: last digit, modulo operation, number theory

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