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If $n$ is a positive integer then the units digit of $66^{n}$ is $6$. So when a power of $66$ is divided by $2$, the units digit of the quotient is either $3$ or $8$. Now $66^{66}$ is clearly a multiple of $4$, so $\frac{1}{2}\left(66^{66}\right)$ is even and therefore has units digit $8$ rather than $3$.

This problem is taken from the UKMT Mathematical Challenges.
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