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What is the smallest number n> 5 so that 5 ^ n ends with "3125"?

What other examples are there?

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    $\begingroup$ What is your take on this? $\endgroup$ – ADITYA PRAKASH Mar 20 at 20:01
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    $\begingroup$ Why not just list them out and find it? $\endgroup$ – Jair Taylor Mar 20 at 20:02
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    $\begingroup$ Why not just do it? It's not $1$ because $5^1=5$. It's not $2$ because $5^2 = 25$. What's to keep you from just continuing? $\endgroup$ – fleablood Mar 20 at 20:20
  • $\begingroup$ The answer to "What is the smallest such n>5?" is easy, so you might as well retitle the question "What are all n>5 such that...?" $\endgroup$ – smci Mar 20 at 23:55
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Hint: $5^n \equiv 5^5 \mod 10^4$ if and only if $5^n \equiv 5^5 \mod 2^4$. What is the multiplicative order of $5$ mod $16$?

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So, we are looking for all $n>5$ for which $5^n\equiv 3125=5^5\mod 10000$.

Note that the following equivalence holds for $n>5$:$$5^n\equiv 5^5\mod 10000\\\iff \\5^{n-4}\equiv 5\mod 16\\\iff\\5^{n-5}\equiv 1\mod 16$$Define $m\triangleq n-5\ge 1$. Then all the $m$s for which $5^{m}\equiv 1\mod 16$ holds are $$m=4k\quad,\quad k\in \Bbb N$$this is because $5^4=625\equiv 1\mod 16$ and therefore $$5^{4k}\equiv5^{4k-4}\equiv\cdots \equiv 5^{4}\equiv 1\mod 16$$

Conclusion

All $n>5$s for which $5^n$ ends up with $3125$ can be found from $$n=4k+5\quad,\quad k\in \Bbb N$$ and the smallest such $n$ is 9.

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Well $$5^9=1953125$$ so the answer is $9$. In fact $$5^n\equiv 5^{n-4} \mod{10^4}$$ For $n\ge 8$, so any value of $5^{5+4k}$ where $k\in\mathbb{N}$ has the last four digits $3125$.

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  • $\begingroup$ Why not $5^5 = 3125$. $\endgroup$ – fleablood Mar 20 at 20:20
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    $\begingroup$ The question states that $n\gt5$ $\endgroup$ – Peter Foreman Mar 20 at 20:22
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Hint $\,\ 5^{\large 5+N}\! \bmod 10^{\large 4} = 5^{\large 5}(5^{\large\color{#c00} N}\! \bmod 2^{\large 4}).\,$ Now recall $\, \begin{align} 5\, &\equiv 1\!\pmod{\! \color{#c00}4} \\ \Rightarrow\ 5^{\large\color{#c00} 4}\!&\equiv 1^{\large\color{#c00} 4}\!\!\!\! \pmod{\!\color{#c00}4^{\large 2}}\end{align}$

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