2005 AMC 12A Problems/Problem 18
Problem
Call a number prime-looking if it is composite but not divisible by or 5. The three smallest prime-looking numbers are
, and
. There are
prime numbers less than
. How many prime-looking numbers are there less than
?
Solution
The given states that there are prime numbers less than
, which is a fact we must somehow utilize. Since there seems to be no easy way to directly calculate the number of "prime-looking" numbers, we can apply complementary counting. We can split the numbers from
to
into several groups:
. Hence, the number of prime-looking numbers is
(note that
are primes).
We can calculate using the Principle of Inclusion-Exclusion: (the values of
and their intersections can be found quite easily)
![$|S_2 \cup S_3 \cup S_5| = |S_2| + |S_3| + |S_5| - |S_2 \cap S_3| - |S_3 \cap S_5| - |S_2 \cap S_5| + |S_2 \cap S_3 \cap S_5|$](http://latex.artofproblemsolving.com/1/7/c/17c5721db444774f1dd96630410388d954efab60.png)
![$= 500 + 333 + 200 - 166 - 66 - 100 + 33 = 734$](http://latex.artofproblemsolving.com/4/9/5/49525cc67578276e81303db6e05daa42698d2833.png)
Substituting, we find that our answer is .
See also
2005 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
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All AMC 12 Problems and Solutions |