1. We need the remainder when $3^{2022}$ is divided by $5$.

2. Find the pattern of powers of $3$ modulo $5$:

   $3^1 \equiv 3 \pmod{5}$ $3^2 = 9 \equiv 4 \pmod{5}$ $3^3 = 27 \equiv 2 \pmod{5}$ $3^4 = 81 \equiv 1 \pmod{5}$

   So the remainders repeat every $4$ powers: $3,4,2,1,3,4,2,1,\dots$

3. Now divide the exponent $2022$ by $4$: $2022 = 4 \times 505 + 2$

   Hence, $3^{2022} = 3^{4\cdot 505 + 2} = (3^4)^{505} \cdot 3^2$

4. Taking modulo $5$: $3^4 \equiv 1 \pmod{5}$ Therefore, $3^{2022} \equiv 1^{505} \cdot 3^2 \equiv 9 \equiv 4 \pmod{5}$

5. So the remainder is $4$.

6. Checking options:

   • A: $1$ ❌
   • B: $2$ ❌
   • C: $3$ ❌
   • D: $4$ ✅

Therefore, the correct option is D.