1. We need the remainder when $7^{2022}+3^{2022}$ is divided by $5$.

So we work modulo $5$.

2. Reduce the bases modulo $5$: $7 \equiv 2 \pmod{5}$ So, $7^{2022} \equiv 2^{2022} \pmod{5}$

Also, $3^{2022} \equiv 3^{2022} \pmod{5}$

Thus we need $2^{2022}+3^{2022} \pmod{5}.$

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

For $2^n$ modulo $5$: $2^1 \equiv 2$ $2^2 \equiv 4$ $2^3 \equiv 8 \equiv 3$ $2^4 \equiv 16 \equiv 1 \pmod{5}$ So the cycle length is $4$.

For $3^n$ modulo $5$: $3^1 \equiv 3$ $3^2 \equiv 9 \equiv 4$ $3^3 \equiv 12 \equiv 2$ $3^4 \equiv 6 \equiv 1 \pmod{5}$ Again, the cycle length is $4$.

4. Now reduce the exponent $2022$ modulo $4$: $2022 \div 4 = 505 \text{ remainder } 2$ So, $2022 \equiv 2 \pmod{4}.$

Hence, $2^{2022} \equiv 2^2 \equiv 4 \pmod{5}$ and $3^{2022} \equiv 3^2 \equiv 9 \equiv 4 \pmod{5}.$

5. Add them: $2^{2022}+3^{2022} \equiv 4+4 = 8 \equiv 3 \pmod{5}.$

Therefore, the remainder is $\boxed{3}.$

6. Checking options:

• A: $0$ ❌
• B: $2$ ❌
• C: $3$ ✅
• D: $4$ ❌

So the correct option is C.