1. We need the remainder when $27^{999}$ is divided by $7$.

2. First reduce the base modulo $7$: $27 \equiv 6 \pmod{7}$ So, $27^{999} \equiv 6^{999} \pmod{7}$

3. Now note that $6 \equiv -1 \pmod{7}$ Hence, $6^{999} \equiv (-1)^{999} \pmod{7}$

4. Since $999$ is odd, $(-1)^{999} = -1$ Therefore, $6^{999} \equiv -1 \pmod{7}$

5. A remainder must be one of $0,1,2,3,4,5,6$. Since $-1 \pmod{7}$ is the same as $6 \pmod{7}$, $27^{999} \equiv 6 \pmod{7}$

6. So the remainder is $6$.

7. Checking options:

• A: $1$ ✗
• B: $2$ ✗
• C: $3$ ✗
• D: $6$ ✓

Therefore, the correct option is D.