### Instruction: What is the ones digit of $7^{35}$ when written as an integer?### Response: 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.