1. **Problem Statement:** Investigate the last digits of integers raised to powers, specifically focusing on the pattern of the last one or two digits. 2. **Understanding the Problem:** When we raise an integer to a power, the last digits often follow a repeating pattern. For example, $7^3 = 343$ ends with the digits 43. 3. **Key Concept:** The last digit of a number raised to powers cycles in a predictable way due to modular arithmetic. For the last digit, we consider modulo 10; for the last two digits, modulo 100. 4. **Example with 7:** - $7^1 = 7$ ends with 7 - $7^2 = 49$ ends with 9 - $7^3 = 343$ ends with 3 - $7^4 = 2401$ ends with 1 - $7^5 = 16807$ ends with 7 (pattern repeats every 4 powers) 5. **General Rule:** The last digit of powers of 7 cycles every 4 steps: 7, 9, 3, 1. 6. **For last two digits:** The pattern is more complex but can be found by calculating powers modulo 100. 7. **Formula:** To find the last $k$ digits of $a^n$, compute $$a^n \mod 10^k$$ 8. **Summary:** The last digits of powers repeat in cycles determined by modular arithmetic, which can be used to predict the ending digits without full calculation.