\(a^2\equiv 0 \pmod{12}\\ \text{That means }\dfrac{a^2}{12}\mbox{ is a positive integer or 0}\\ \mbox{However }a\neq 0 \mbox{ so }\dfrac{a^2}{12} \mbox{ can't be 0.}\\ \begin{array}{rcl}\\&&a^2=12x\end{array}\\ \text{Check the multiples of 12 from 12 to 144}\\ \text{12 is not a square number}\\\text{24 is not a square number}\\\color{red}\text{36 is a square number} \\\text{48 is not a square number}\\ \)
\(\text{60 is not a square number}\\\text{72 is not a square number}\\\text{84 is not a square number}\\\text{96 is not a square number} \\\text{108 is not a square number}\\\text{120 is not a square number}\\\text{132 is not a square number}\\ \text{144 is not a square number}\)
\(\begin{array}{rllll}&a^2&=&36&\\&a&=&\pm 6&\\&\text{a must not be -6}\\\therefore&a=6.\end{array}\)
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