Mock AIME 1 Pre 2005 Problems/Problem 4

Problem

When $1 + 7 + 7^2 + \cdots + 7^{2004}$ is divided by $1000$ , a remainder of $N$ is obtained. Determine the value of $N$ .

Solution

By the geometric series formula, $1 + 7 + 7^2 + \cdots + 7^{2004} = \frac{7^{2005}-1}{7-1} = \frac{7^{2005}-1}{6}$ . Since $\varphi(1000) = 400$ , by Fermat-Euler's Theorem, this is equivalent to finding $\frac{7^{400 \cdot 5 + 6} - 1}{6} \equiv \frac{7^5 - 1}{6} \equiv \boxed{801} \pmod{1000}$ .

Mock AIME 1 Pre 2005 (Problems, Source)
Preceded by Problem 3	Followed by Problem 5
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Mock AIME 1 Pre 2005 Problems/Problem 4

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