If $p$ is a prime number greater than $3$, what is the remainder when $p^2$ is divided by $12$?

Step 1 (Modulo 12 residues): Any prime number $p>3$ must be coprime to both 2 and 3. Therefore, $p$ must be coprime to 12. The possible residues of $p(\mathrm{mod}12)$ are:

Step 2 (Square the residues): We square each possible residue class mod 12:

• $1^2=1\equiv 1(\mathrm{mod}12)$

• $5^2=25\equiv 1(\mathrm{mod}12)$

• $7^2=49\equiv 1(\mathrm{mod}12)$

• $11^2=121=10\times 12+1\equiv 1(\mathrm{mod}12)$

Step 3 (Conclusion): For any prime $p>3$, $p^2$ always leaves a remainder of $1$ when divided by 12. This corresponds to option index 0 in ['1', '5', '7', '11'].