2005 AMC 12A Problems/Problem 18
Problem
Call a number "prime-looking" if it is composite but not divisible by 
, 
, or 
. 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)
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 | |
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