Difference between revisions of "2002 AMC 12A Problems/Problem 6"
(New page: ==Problem== For how many positive integers <math>m</math> does there exist at least one positive integer n such that <math>m \cdot n \le m + n</math>? <math> \mathrm{(A) \ } 4\qquad \math...) |
|||
Line 1: | Line 1: | ||
+ | {{duplicate|[[2002 AMC 12A Problems|2009 AMC 12A #6]] and [[2002 AMC 10A Problems|2009 AMC 10A #4]]}} | ||
+ | |||
==Problem== | ==Problem== | ||
For how many positive integers <math>m</math> does there exist at least one positive integer n such that <math>m \cdot n \le m + n</math>? | For how many positive integers <math>m</math> does there exist at least one positive integer n such that <math>m \cdot n \le m + n</math>? | ||
Line 6: | Line 8: | ||
==Solution== | ==Solution== | ||
− | |||
− | <math>m \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>, |
+ | therefore the answer is <math>\boxed{\text{(E) infinitely many}}</math>. | ||
+ | |||
+ | ==See Also== | ||
+ | |||
+ | {{AMC12 box|year=2002|ab=A|num-b=5|num-a=7}} | ||
+ | {{AMC10 box|year=2002|ab=A|num-b=3|num-a=5}} | ||
− | + | [[Category:Introductory Algebra Problems]] |
Revision as of 07:56, 18 February 2009
- The following problem is from both the 2009 AMC 12A #6 and 2009 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
For any we can pick
, we get
,
therefore 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 |