Here's how I view this....
5^1 / 8 = (8 - 3)^1 / 8
Every term in the expansion is a multiple of 8 except the last which is
(-3)^1 / 8 = (-3)^1 mod 8 = 5
Similarly
5^2 /8 = (8 - 3)^2 / 8
Again we only need consider the last term which is
(3)^2/8 = (3)^2 mod 8 = 1
etc....
So...we only need to consider the last term in each subsequent power expansion and note that
(3)^n mod 8 = 1 and
(-3)^n mod 8 = 5
So
5^137 / 8 = (8 - 3)^137 / 8 will have (-3)^137 / 8 as a last term
And the remainder is (-3)^137 mod 8 = 5
