Difference between revisions of "2002 AMC 12A Problems/Problem 6"
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<math> \mathrm{(A) \ } 4\qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 9\qquad \mathrm{(D) \ } 12\qquad \mathrm{(E) \ }</math> infinitely many | <math> \mathrm{(A) \ } 4\qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 9\qquad \mathrm{(D) \ } 12\qquad \mathrm{(E) \ }</math> infinitely many | ||
− | + | ==Solution== | |
− | ==Solution 1== | + | ===Solution 1=== |
For any <math>m</math> we can pick <math>n=1</math>, we get <math>m \cdot 1 \le m + 1</math>, | For any <math>m</math> we can pick <math>n=1</math>, we get <math>m \cdot 1 \le m + 1</math>, | ||
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− | ==Solution 2== | + | ===Solution 2=== |
− | Another solution, slightly similar to this first one would be using Simon's Favorite Factoring Trick. | + | Another solution, slightly similar to this first one would be using [[Simon's Favorite Factoring Trick]]. |
<math>(m-1)(n-1) \leq 1</math> | <math>(m-1)(n-1) \leq 1</math> |
Revision as of 15:36, 29 July 2011
- The following problem is from both the 2002 AMC 12A #6 and 2002 AMC 10A #4, so both problems redirect to this page.
Contents
[hide]Problem
For how many positive integers does there exist at least one positive integer n such that ?
infinitely many
Solution
Solution 1
For any we can pick , we get , therefore the answer is .
Solution 2
Another solution, slightly similar to this first one would be using Simon's Favorite Factoring Trick.
Let , then
This means that there are infinitely many numbers that can satisfy the inequality. So the answer is .
See Also
2002 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 5 |
Followed by Problem 7 |
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 |
2002 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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 |