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 :)

  • $\begingroup$ 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

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 $4$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.