Test every square

Only possible modulo 6 residues are 0, 1, 2, 3, 4 and 5. On squaring them,

${ 0 }^{ 2 }\equiv 0\quad \left( \text{mod 6} \right) \\ { 1 }^{ 2 }\equiv 1\quad \left( \text{mod 6} \right) \\ { 2 }^{ 2 }\equiv 4\quad \left( \text{mod 6} \right) \\ { 3 }^{ 2 }\equiv 3\quad \left( \text{mod 6} \right) \\ { 4 }^{ 2 }\equiv 4\quad \left( \text{mod 6} \right) \\ { 5 }^{ 2 }\equiv 1\quad \left( \text{mod 6} \right)$

Since 2 doesn't show up, the answer is 2.