Difference between revisions of "2001 AIME II Problems/Problem 10"
(fix) |
|||
Line 14: | Line 14: | ||
[[Category:Intermediate Number Theory Problems]] | [[Category:Intermediate Number Theory Problems]] | ||
+ | {{MAA Notice}} |
Revision as of 20:35, 4 July 2013
Problem
How many positive integer multiples of can be expressed in the form
, where
and
are integers and
?
Solution
The prime factorization of . We have
. Since
, we require that
. From the factorization
, we see that
works; also,
implies that
, and so any
will work.
To show that no other possibilities work, suppose , and let
. Then we can write
, and we can easily verify that
for
.
If , then we can have solutions of
ways. If
, we can have the solutions of
, and so forth. Therefore, the answer is
.
See also
2001 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.