Difference between revisions of "2005 AMC 12A Problems/Problem 18"
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Revision as of 19:58, 20 June 2020
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 |
| 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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
