If $3^2019$ is divided by 10, then what is the remainder?

Correct Answer: (C) 7

Dividing a positive integer by 10 yields its units digit. The units digits of powers of 3 cycle with period 4: $3^1$$\equiv$ 3, 3^2 $\equiv$ 9, 3^3 $\equiv$ 7, 3^4 $\equiv$ 1. Dividing the exponent by 4 gives $2019 = 4$$\times$ 504 + 3 $\equiv$ 3 \pmod 4. Thus, the units digit of $3^{2019}$ is the same as that of $3^3$, which is 7.