# Last four digits of $S=9+9^2+9^3+\ldots+9^{400}$

What are the last four digits of the following sum:

$$S=9+9^2+9^3+\ldots+9^{400} ?$$

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No attempt at the answer?? – Epictetus Dec 3 '12 at 10:58

Hint: This is a geometric sum, you can sum it and then do mod $10000$ to get the last $4$ digits
HINT 2: Use Euler's Theorem: if $a$ and $m$ are coprime, we have $a^n\equiv a^{n \text{ mod }\varphi(m)} \text{ mod }m$, where $\varphi$ is Euler's totient function. What is $\varphi(10000)$?