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Solution

We use modulo properties: $27 \equiv 3 \pmod{12}$ $\Rightarrow 27^{40} \equiv 3^{40} \pmod{12}$ Now, $3^1 \equiv 3$, $3^2 \equiv 9$, $3^3 \equiv 3$, $3^4 \equiv 9$, pattern repeats every 2 powers. So, for even power $40$, remainder = $9$.