2001 AMC 12 Problems/Problem 12
- The following problem is from both the 2001 AMC 12 #12 and 2001 AMC 10 #25, so both problems redirect to this page.
Problem
How many positive integers not exceeding 
are multiples of 
or 
but not 
?
Solution
Out of the numbers 
to 
four are divisible by and three by , counting twice.
Hence 
out of these numbers are multiples of or .
The same is obviously true for the numbers 
to 
for any positive integer 
.
Hence out of the numbers to 
there are 
numbers that are divisible by or .
Out of these 
, the numbers 
, 
, , 
, 
and 
are divisible by .
Therefore in the set 
there are precisely 
numbers that satisfy all criteria from the problem statement.
Again, the same is obviously true for the set 
for any positive integer .
We have 
, hence there are 
good numbers among the numbers to 
. At this point we already know that the only answer that is still possible is 
, as we only have numbers left.
By examining the remaining by hand we can easily find out that exactly 
of them match all the criteria, giving us 
good numbers.
This is correct.
See Also
| 2001 AMC 12 (Problems • Answer Key • Resources) | |
| Preceded by Problem 11 |
Followed by Problem 13 |
| 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 | |
| 2001 AMC 10 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 24 |
Followed by Last Question | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
