Difference between revisions of "2007 AMC 10B Problems/Problem 25"
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Revision as of 11:23, 4 July 2013
How many pairs of positive integers (a,b) are there such that a and b have no common factors greater than 1 and:
is an integer?
Solution
Let 
. We can then write the given expression as 
where 
is an integer. We can rewrite this as a quadratic, 
. By the Quadratic Formula, 
. We know that 
must be rational, so 
must be a perfect square. Let 
. Then, 
. The factors pairs of 
are 
and , 
and 
, 
and 
, and 
and 
. Only and and and give integer solutions, 
and 
and 
and 
, respectively. Plugging these back into the original equation, we get 
possibilities for , namely 
and 
.
See Also
| 2007 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 24 |
Followed by Last question | |
| 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 | ||
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