Remainder?

Note first that $3^{3} = 27 = 4 \times 7 - 1 \equiv -1 \pmod{7}$ and that $1989 = 3 \times 663$ . Thus

$3^{1989} = (3^{3})^{663} \equiv (-1)^{663} \pmod{7} \equiv -1 \pmod{7} \equiv \boxed{6} \pmod{7}$ .

Relevant wiki: Euler's Theorem

$\begin{aligned} 3^{1989} \pmod 7 &= 3^{1989 \pmod{\phi (7)}} \pmod 7 & \small{\color{#3D99F6} \text{As } \text{gcd}(3,7) = 1 \text{ we can apply Euler's theorem}} \\ &= 3^3 \pmod 7 \\ & \equiv \boxed{6} \end{aligned}$