2003 AMC 12A Problems/Problem 8
- The following problem is from both the 2003 AMC 12A #8 and 2003 AMC 10A #8, so both problems redirect to this page.
Problem
What is the probability that a randomly drawn positive factor of is less than ?
Solution
Solution 1
For a positive number which is not a perfect square, exactly half of the positive factors will be less than .
Since is not a perfect square, half of the positive factors of will be less than .
Clearly, there are no positive factors of between and .
Therefore half of the positive factors will be less than .
So the answer is .
Solution 2
Testing all numbers less than , numbers , and divide . The prime factorization of is . Using the formula for the number of divisors, the total number of divisors of is . Therefore, our desired probability is
Solution 3
This is not too bad with casework. Notice that . Hence, has factors, of which are less than . Thus, the answer is .
Solution by franzliszt
See Also
2003 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2003 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 7 |
Followed by Problem 9 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.