Let (13!)^4 = x
(x^4 - x^2)/ (x^2 - x) = a
by solving further we get:
x^3 + x^2 = ax
or a/x = x + 1
a/(13! ^ 4 ) = (13!^ 4) + 1

as 13! ends with 0, so is 13!^ 4; so (13!^ 4) + 1 ends with 1.
Unit's digit is 1.