Mock AIME 2 Pre 2005 Problems/Problem 4

Problem

Let $S = \{5^k | k \in \mathbb{Z}, 0 \le k \le 2004 \}$ . Given that $5^{2004} = 5443 \cdots 0625$ has $1401$ digits, how many elements of $S$ begin with the digit $1$ ?

Solution

Note that $5^n$ has the same number of digits as $5^{n-1}$ if and only if $5^{n-1}$ has a leading digit $1$ . Therefore, there are $2004 - 1401 = 603$ numbers with leading digit $1$ among the set $\{5^1, 5^2, 5^3, \cdots 5^{2003}\}.$ However, $5^0$ also starts with $1$ , so the answer is $603 + 1 = \boxed{604}$ .

-MP8148

Mock AIME 2 Pre 2005 (Problems, Source)
Preceded by Problem 3	Followed by Problem 5
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Mock AIME 2 Pre 2005 Problems/Problem 4

Problem

Solution

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