You need to use some properties of modular arithmetic.

basically (a * a) mod b = (a mod b) * (a mod b)

therefore, you can figure out the remainder of 7^n divided by 8 if you know the remainder of 7^(n-1) divided by 8

Thus since you know 7^2 (mod 8) = 1 then 7^3 (mod 8) = 7^2(mod 8) * 7(mod 8) = 7

in these types of problems, look for cyles

remainder when (7^1) / 8 : 7
remainder when (7^2) / 8 : 1
remainder when (7^3) / 8 : 7
remainder when (7^4) / 8 : 1
remainder when (7^5) / 8 : 7

Therefore we see when the power is odd, the remainder is 7, so thats your answer.