5^an constant mod 100000?
Number Theory
Level
5
\[ \begin{array} { l c c }
25 ^ 1 & \equiv 25 & \pmod{1000} \\
25 ^ 2 & \equiv 625 & \pmod{1000} \\
25 ^ 3 & \equiv 625 & \pmod{1000} \\
25 ^ 4 & \equiv 625 & \pmod{1000} \\
\vdots & \vdots & \vdots \\
\end{array} \]
We know that the last 3 digits of \( 25 ^ n \) will always be a constant 625 for large enough integer values of \( n \).
What is the smallest positive integer value \(a\) such that the last 5 digits of \( \left( 5 ^ a \right) ^n \) will always be a constant for large enough integer values of \( n \)?
by
Calvin Lin