lean_workbook_plus_12475

Now, we want to find $2^{30} \pmod{1000}$ . All congruences here will be in $\pmod{1000}$ . We note that $2^{10} = 1024 \equiv 24$ . Now our problem has turned into $24^3$ . There may be some clever way to evaluate this, but it isn't that tedious to just multiply out, knowing that $24^2 = 576$ . We get $24^3 = 13824 \equiv \boxed{824}$