Chapter 1: Understanding Modulo Arithmetic

Finding the last digit of a massive number can be a daunting task, especially when it consists of 846 digits! Fortunately, there are efficient methods to tackle this problem using just a pen and paper, primarily through the concept of modulo arithmetic.

Modulo arithmetic is similar to the way we tell time on a clock. For a given integer m, we say two integers are 'congruent' modulo m (symbolized by a triple equals sign) if their difference is a multiple of m. For instance, the expression 17 ≡ 5 (mod 12) reflects the idea that 5 am and 5 pm refer to the same hour on a clock.

By applying modulo arithmetic, we can simplify an integer to its smallest positive congruence, making calculations easier. For example, we can find 172 ≡ 7 (mod 15) and 137 ≡ 2 (mod 15). Therefore, to calculate 172 * 137 (mod 15), we can simply multiply their congruences: 172 * 137 ≡ 7 * 2 ≡ 14 (mod 15).

Next, let’s apply this to our specific challenge. Determining the last digit of an integer is equivalent to finding its smallest positive congruence modulo 10.

7 ≡ 7 (mod 10).

Here, we can utilize negative numbers to derive more useful congruences. Notably,

9 ≡ -1 (mod 10),

This insight allows us to perform further calculations more effortlessly.

Since 1 raised to any positive integer remains 1, we can conclude that 7 raised to any positive integer power will still be congruent to 1 modulo 10. For instance, we can express 1000 as 250 * 4, leading us to complete our calculation:

Consequently, the last digit of (7^{1000}) is determined to be 1.

Extension: Challenge yourself to find the last digit of (7^{2021}). A hint: 7³ ≡ 3 (mod 10) suggests that the last digits of powers of 7 follow a cyclic pattern of 7, 9, 3, 1.

Stay tuned for an upcoming, more complex example!

Chapter 2: Practical Applications of Modulo Arithmetic

In this chapter, we will dive deeper into practical applications of modulo arithmetic techniques to solve various mathematical problems and puzzles.