Last digit (1)

$1999 \mod 4 = 3 \implies \ 93^3 \mod 10 = 7$ because we need the last digit, we consider mod 10.

We only need to consider the units' digit of the base number and the remainder yielded when the exponent is divided by 4. If the remainder is 1, 2, 3, or 0, then the units digit of the number (when using the units' digit 3) is 3, 9, 7, or 1 respectively. In this case 1999/4 yields a remainder of 3, so the units' digit of the number 93 to the power of 1999 is equal to $\boxed{7}$ .