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Difference between revisions of "2012 AIME I Problems/Problem 15"

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==Problem 15==
 
==Problem 15==
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There are <math>n</math> mathematicians seated around a circular table with <math>n</math> seats numbered <math>1,</math> <math>2,</math> <math>3,</math> <math>...,</math> <math>n</math> in clockwise order. After a break the again sit around the table. The mathematicians note that there is a positive integer <math>a</math> such that
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<UL>
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(<math>1</math>) for each <math>k,</math> the mathematician who was seated in seat <math>k</math> before the break is seated in seat <math>ka</math> after the break (where seat <math>i + n</math> is seat <math>i</math>);
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</UL>
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<UL>
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(<math>2</math>) for every pair of mathematicians, the number of mathematicians sitting between them after the break, counting in both the clockwise and the counterclockwise directions, is different from either of the number of mathematicians sitting between them before the break.
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</UL>
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Find the number of possible values of <math>n</math> with <math>1 < n < 1000.</math>
  
 
== Solution ==
 
== Solution ==

Revision as of 01:47, 17 March 2012

Problem 15

There are $n$ mathematicians seated around a circular table with $n$ seats numbered $1,$ $2,$ $3,$ $...,$ $n$ in clockwise order. After a break the again sit around the table. The mathematicians note that there is a positive integer $a$ such that

    ($1$) for each $k,$ the mathematician who was seated in seat $k$ before the break is seated in seat $ka$ after the break (where seat $i + n$ is seat $i$);
    ($2$) for every pair of mathematicians, the number of mathematicians sitting between them after the break, counting in both the clockwise and the counterclockwise directions, is different from either of the number of mathematicians sitting between them before the break.

Find the number of possible values of $n$ with $1 < n < 1000.$

Solution

See also

2012 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Last Problem
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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