Difference between revisions of "2003 AMC 10A Problems/Problem 24"
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[[Category:Introductory Number Theory Problems]] | [[Category:Introductory Number Theory Problems]] |
Revision as of 10:16, 15 January 2008
Problem
Sally has five red cards numbered through
and four blue cards numbered
through
. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards?
Solution
Let and
designate the red card numbered
and the blue card numbered
, respectively.
is the only blue card that
evenly divides, so
must be at one end of the stack and
must be the card next to it.
is the only other red card that evenly divides
, so
must be the other card next to
.
is the only blue card that
evenly divides, so
must be at the other end of the stack and
must be the card next to it.
is the only other red card that evenly divides
, so
must be the other card next to
.
doesn't evenly divide
, so
must be next to
,
must be next to
, and
must be in the middle.
This yields the following arrangement from top to bottom:
Therefore, the sum of the numbers on the middle three cards is .
See Also
2003 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |