Twenty-three sevens are multiplied together: $7\times 7\times 7\times$ … × 7. What is the last digit of the product?

Track only the last digit: 7, then 49 → 9, then 343 → 3, then 2401 → 1, and after that the pattern 7, 9, 3, 1 repeats every four steps. Since $23=4\times 5+3$, the twenty-third seven lands on the third place of the cycle, which is 3. The traps: 7 assumes the last digit never changes, 9 comes from a remainder of 2, and 1 from a remainder of 4.