Difference between revisions of "2001 AMC 12 Problems/Problem 21"
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Revision as of 19:58, 18 February 2015
Problem
Four positive integers ,
,
, and
have a product of
and satisfy:
What is ?
Solution
Using Simon's Favorite Factoring Trick, we can rewrite the three equations as follows:
Let . We get:
Clearly divides
. On the other hand,
can not divide
, as it then would divide
. Similarly,
can not divide
. Hence
divides both
and
. This leaves us with only two cases:
and
.
The first case solves to , which gives us
, but then
. (We do not need to multiply, it is enough to note e.g. that the left hand side is not divisible by
.)
The second case solves to , which gives us a valid quadruple
, and we have
.
See Also
2001 AMC 12 (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
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 12 Problems and Solutions |
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