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.
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 | ||
