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How many factors of $2010^{2010}$ have last digit $2$?

It is not difficult to solve this using python/mathematica, but I want to know how to (smartly) solve this one with paper and pencil?

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    Preliminary question: are you familiar with [modular arithmetic](http://en.wikipedia.org/wiki/Modular_arithmetic)?2011-11-03
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    The problem would be easier if the exponent was $2007$, because the digits of $3^a$ and $67^b$ each cycle with period $4$. For any of $2$, $4$, $8$, $6$ there are four patterns $(a,b)$ within each cycle that multiplied by $2$ yield $2$, also four that multiplied by $4$ yield $2$, and so on. There is an additional bit of asymmetry because the digits of $2^c$ cycle with period $4$, with the exceptional $1$ at $c=0$.2011-11-03

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