Difference between revisions of "2001 AMC 12 Problems/Problem 12"
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Revision as of 21:03, 3 July 2013
- 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.
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.