Difference between revisions of "2013 AMC 10A Problems/Problem 19"
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− | We want the integers <math>b</math> such that <math> 2013\equiv 3\pmod{b} \Rightarrow b </math> is a factor of <math>2010</math>. Since <math>2010=2 \cdot 3 \cdot 5 \cdot 67</math>, it has <math>(1+1)(1+1)(1+1)(1+1)=16</math> factors. Since <math>b</math> cannot equal <math>1, 2, </math> or <math>3</math>, our answer is <math>16-3=\boxed{\textbf{(C) }13 | + | We want the integers <math>b</math> such that <math> 2013\equiv 3\pmod{b} \Rightarrow b </math> is a factor of <math>2010</math>. Since <math>2010=2 \cdot 3 \cdot 5 \cdot 67</math>, it has <math>(1+1)(1+1)(1+1)(1+1)=16</math> factors. Since <math>b</math> cannot equal <math>1, 2, </math> or <math>3</math>, our answer is <math>16-3=\boxed{\textbf{(C) }13}</math> |
Revision as of 21:20, 7 February 2013
Problem
In base , the number ends in the digit . In base , on the other hand, the same number is written as and ends in the digit . For how many digits does the base--representation of end in the digit ?
Solution
We want the integers such that is a factor of . Since , it has factors. Since cannot equal or , our answer is