What is the last digit of $7^{35}$?

Explanation

1. 01

   The last digit is the remainder modulo 10. The last digits of $7, 7^{2}, 7^{3}, 7^{4}$ are 7, 9, 3, 1, and then they repeat.

2. 02

   $35 = 4 \times 8 + 3$, so $7^{35}$ ends like $7^{3} = 343$: in 3.