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In a math competition I came across the following question:

What digit does the result of 2^2006 end with?

This competition tested how fast you are at solving math problems. So, I was wondering whether there is some sort of shortcut to solve problems like this quickly.

Help would be appreciated.

Thank you :)

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  • $\begingroup$ End digits of powers of $2$: $2,4,8,16=6,12=2,4,8,...$. $\endgroup$ – String Apr 16 '15 at 8:53
  • $\begingroup$ Thus if $n\equiv m\pmod{4}$ we have $2^n\equiv 2^m\pmod{10}$. Now $2006=4\cdot 501+2\equiv 2\pmod{4}$. So the last digit of $2^{2006}$ is the same as the last digit of $2^2$, which happens to be $4$. $\endgroup$ – String Apr 16 '15 at 8:58
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Notice the pattern among last digits of ascending powers of $2$

Last digits of:

$2^1$ is $2$, $2^2$ is $4$, $2^3$ is $8$, $2^4$ is $6$, $2^5$ is $2$, $2^6$ is $4...$ etc

Also notice that the last digits of

$2^4$, $2^8$, $2^{12}$, $2^{16}$... will all be $6$

We note that as $2004$ is also a multiple of $4$, therefore $2^{2004}$ will have a last digit of $6$

Continuing the pattern $2^{2005}$ will have a last digit of $2$

And $2^{2006}$ has a last digit of $4$

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