jmc

For this problem, we do not actually need to multiply out the whole thing, and find out exactly what ${\left(\frac{1}{23}\right)}^{1000}$ equals. That would take a long time! Instead, realize that when we multiply a small fraction like this by itself, the result is smaller than our original fraction. Then if we multiply that result by the original fraction again, the result will be even smaller. Now imagine if we do this $1000$ times. The final result will be a $\text{very}$ small fraction, close to the integer $\boxed{0}$.