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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