I just want to check that I did this right. So for powers of $7$, they have a pattern of: $$7^1 = 7, \; 7^2 = 9, \; 7^3 = 3, \; 7^4 = 1, \; 7^5 = 7$$
So for $3553$, I divide it by $4$ and get $888.25$. So it has a remainder of 1/4 which means the last digit is the beginning of a new cycle, so it is $7$.