Advent calendar 2022
4 December
The last three digits of \(5^5\) are 125.
What are the last three digits of \(5^{2,022,000,000}\)?
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The last three digits of \(5^1\) to \(5^7\) are:
$$
\begin{align*}
5^1&=5&&\rightarrow&&5\\
5^2&=25&&\rightarrow&&25\\
5^3&=125&&\rightarrow&&125\\
5^4&=625&&\rightarrow&&625\\
5^5&=3125&&\rightarrow&&125\\
5^6&=15625&&\rightarrow&&625\\
5^7&=78125&&\rightarrow&&125\\
\end{align*}
$$
This pattern continues: the last three digits of 5 to the power of any odd number (greater than 2) is 125, and the last three digits of 5 to the power of any even number (greater than 3) is 625.
The last three digits of \(5^{2,022,000,000}\) are therefore 625.
