2. [3] What is the units digit of $7^{2009}$ ?
Answer: 7 Note that the units digits of $7^{1}, 7^{2}, 7^{3}, 7^{4}, 7^{5}, 7^{6}, \ldots$ follows the pattern $7,9,3,1,7,9,3,1, \ldots$.. The 2009th term in this sequence should be 7 .
Alternate method: Note that the units digit of $7^{4}$ is equal to 1 , so the units digit of $\left(7^{4}\right)^{502}$ is also 1. But $\left(7^{4}\right)^{502}=7^{2008}$, so the units digit of $7^{2008}$ is 1 , and therefore the units digit of $7^{2009}$ is 7 .