What is the ones digit of $7^{35}$ when written as an integer?
To find the ones digit of a large power of 7, I can look at the pattern of the ones digits of smaller powers of 7. For example, $7^1 = 7$, $7^2 = 49$, $7^3 = 343$, $7^4 = 2401$, and so on. I notice that the ones digits repeat in a cycle of four: 7, 9, 3, 1, 7, 9, 3, 1, ... So, to find the ones digit of $7^{35}$, I just need to figure out where 35 falls in this cycle. I can use division and remainder to do this. If I divide 35 by 4, I get 8 with a remainder of 3. This means that 35 is 3 more than a multiple of 4, so it is in the third position of the cycle. Therefore, the ones digit of $7^{35}$ is the same as the ones digit of $7^3$, which is 3. 