A7. What are the last four digits of $5^{2013}$?
A) 0625
B) 2525
C) 3125
D) 5625
E) 8125
A7. C) 3125

We look at the last four digits of the powers of 5:

![](https://cdn.mathpix.com/cropped/2024_04_17_88e9b1bd823632ad9c59g-2.jpg?height=185&width=214&top_left_y=450&top_left_x=1521)

$$
\begin{aligned}
& 5^{1}=0005 \quad 5^{5}=3125 \\
& 5^{2}=0025 \quad 5^{6}=15625 \\
& 5^{3}=0125 \quad 5^{7}=78125 \\
& 5^{4}=0625 \quad 5^{8}=390625
\end{aligned}
$$

When calculating the last four digits of a power of 5, it is sufficient to know the last four digits of the previous power of 5. Thus, the last four digits of \(5 \times 390625\) and \(5 \times 0625\) are the same, namely 3125. Since the last four digits of \(5^{4}\) and \(5^{8}\) are the same, the last four digits of the powers of 5 will repeat every four steps from there. The last four digits of \(5^{2013}\) are therefore the same as those of \(5^{2009}\) and \(5^{2005}\), and so on, down to those of \(5^{5}=3125\). The last four digits are thus 3125.