To find the last digit of $3^{2023}$, we need to evaluate $3^{2023} \mod 10$, as the last digit of a number is its remainder when divided by 10.

Step-by-Step Solution:

1. Identify the pattern of the last digits of powers of 3: We can find a repeating cycle for the last digits of powers of 3 by calculating the first few powers of 3 modulo 10: $3^1 = 3 \quad \Rightarrow \quad 3 \mod 10 = 3$ $3^2 = 9 \quad \Rightarrow \quad 9 \mod 10 = 9$ $3^3 = 27 \quad \Rightarrow \quad 27 \mod 10 = 7$ $3^4 = 81 \quad \Rightarrow \quad 81 \mod 10 = 1$ After this, the cycle repeats: $3, 9, 7, 1$.

2. Determine the position in the cycle: The powers of 3 follow a cycle of length 4: $3, 9, 7, 1$. To find the last digit of $3^{2023}$, we need to determine the remainder when 2023 is divided by 4: $2023 \div 4 = 505 \text{ remainder } 3$ So, $3^{2023}$ corresponds to the third number in the cycle, which is 7.

Thus, the last digit of $3^{2023}$ is $\boxed{7}$.

Do you have any questions or would you like further details on any step?

Here are 5 related questions:

1. What is the last digit of $3^{2024}$?
2. How would you find the last digit of $7^{2023}$?
3. What is the sum of the first 10 powers of 3?
4. Can you generalize the last digit pattern for powers of any number?
5. How does modular arithmetic simplify large exponent calculations?

Tip: When working with powers and moduli, identifying repeating cycles helps solve problems efficiently!