Difference between revisions of "Mock AIME 2 2006-2007 Problems/Problem 3"
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== Problem == | == Problem == | ||
− | Let <math> | + | Let <math>S</math> be the sum of all [[positive integer]]s <math>n</math> such that <math>n^2+12n-2007</math> is a [[perfect square]]. Find the [[remainder]] when <math>S</math> is divided by <math>1000.</math> |
==Solution== | ==Solution== | ||
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− | *[[Mock AIME 2 2006-2007/Problem 2 | Previous Problem]] | + | *[[Mock AIME 2 2006-2007 Problems/Problem 2 | Previous Problem]] |
− | *[[Mock AIME 2 2006-2007/Problem 4 | Next Problem]] | + | *[[Mock AIME 2 2006-2007 Problems/Problem 4 | Next Problem]] |
*[[Mock AIME 2 2006-2007]] | *[[Mock AIME 2 2006-2007]] | ||
[[Category:Intermediate Number Theory Problems]] | [[Category:Intermediate Number Theory Problems]] |
Revision as of 14:35, 3 April 2012
Problem
Let be the sum of all positive integers
such that
is a perfect square. Find the remainder when
is divided by
Solution
If , we can complete the square on the left-hand side to get
so
. Subtracting
and factoring the left-hand side, we get
.
, which can be split into two factors in 3 ways,
. This gives us three pairs of equations to solve for
:
and
give
and
.
and
give
and
.
and
give
and
.
Finally, , so the answer is
.