Level
pending

Compute the remainder when $2014^{2014}$ is divided by $1000$ .

The answer is 16.

**
This section requires Javascript.
**

You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.

I'm sure there's an easier way to do this, but when in doubt, use exponentiation by squaring:

$\begin{aligned} 14^{2014}&\equiv196^{1007}\\&\equiv196^{1006}(196)\\&\equiv416^{503}(196)\\&\equiv416^{502}(536)\\&\equiv56^{251}(536)\\&\equiv56^{250}(16)\\&\equiv136^{125}(16)\\&\equiv136^{124}(176)\\&\equiv496^{62}(176)\\&\equiv16^{31}(176)\\&\equiv16^{30}(816)\\&\equiv256^{15}(816)\\&\equiv256^{14}(896)\\&\equiv536^7(896)\\&\equiv536^6(256)\\&\equiv296^3(256)\\&\equiv296^2(776)\\&\equiv616(776)\\&\equiv\boxed{16}\pmod{1000} \end{aligned}$