2001 AIME II Problems/Problem 10
From AoPSWiki
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 • Resources) | ||
| Preceded by Problem 9 | Followed by Problem 11 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
