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What are the last four digits of the following sum:

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

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4  
No attempt at the answer?? –  Epictetus Dec 3 '12 at 10:58

2 Answers 2

up vote 7 down vote accepted

Hint: This is a geometric sum, you can sum it and then do mod $10000$ to get the last $4$ digits

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mod 10000? [spacefiller] –  akkkk Dec 3 '12 at 11:07
    
@akkkk - thanks, updated –  Belgi Dec 3 '12 at 11:17

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 $\varphi(10000)$?

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