2001 AIME II Problems/Problem 10
Contents
Problem
How many positive integer multiples of can be expressed in the form , where and are integers and ?
Solution 1
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 .
Solution 2
Observation: We see that there is a pattern with .
So, this pattern repeats every 6.
Also, , so , and thus, . Continue with the 2nd paragraph of solution 1, and we get the answer of
-AlexLikeMath
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 |
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