Difference between revisions of "1999 AIME Problems/Problem 7"
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== Problem == | == Problem == | ||
| + | There is a set of 1000 switches, each of which has four positions, called <math>A, B, C</math>, and <math>D</math>. When the position of any switch changes, it is only from <math>A</math> to <math>B</math>, from <math>B</math> to <math>C</math>, from <math>C</math> to <math>D</math>, or from <math>D</math> to <math>A</math>. Initially each switch is in position <math>A</math>. The switches are labeled with the 1000 different integers <math>(2^{x})(3^{y})(5^{z})</math>, where <math>x, y</math>, and <math>z</math> take on the values <math>0, 1, \ldots, 9</math>. At step i of a 1000-step process, the <math>i</math>-th switch is advanced one step, and so are all the other switches whose labels divide the label on the <math>i</math>-th switch. After step 1000 has been completed, how many switches will be in position <math>A</math>? | ||
== Solution == | == Solution == | ||
Revision as of 16:29, 12 August 2006
Problem
There is a set of 1000 switches, each of which has four positions, called 
, and 
. When the position of any switch changes, it is only from 
to 
, from to 
, from to , or from to . Initially each switch is in position . The switches are labeled with the 1000 different integers 
, where 
, and 
take on the values 
. At step i of a 1000-step process, the 
-th switch is advanced one step, and so are all the other switches whose labels divide the label on the -th switch. After step 1000 has been completed, how many switches will be in position ?
