Mock AIME 2 2006-2007 Problems/Problem 3
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
.
See Also
Mock AIME 2 2006-2007 (Problems, Source) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |