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2006 Alabama ARML TST Problems/Problem 11

Revision as of 11:13, 29 September 2008 by 1=2 (talk | contribs) (New page: ==Problem== The integer <math>5^{2006}</math> has 1403 digits, and 1 is its first digit (farthest to the left). For how many integers <math>0\leq k \leq 2005</math> does <math>5^k</math> b...)
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Problem

The integer $5^{2006}$ has 1403 digits, and 1 is its first digit (farthest to the left). For how many integers $0\leq k \leq 2005$ does $5^k$ begin with the digit 1?

Solution

Now either $5^k$ starts with 1, or $5^{k+1}$ has one more digit than $5^k$. From $5^0$ to $5^5^2005$ (Error compiling LaTeX. Unknown error_msg), we have 1401 changes, so those must not begin with the digit 1. $2006-1401=\boxed{605}$

See also

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