Problem:
Call a number "prime-looking" if it is composite but not divisible by 2,3 , or 5. The three smallest prime-looking numbers are 49, 77, and 91 . There are 168 prime numbers less than 1000 . How many prime-looking numbers are there less than
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Solution:
Of the numbers less than 1000, 499 of them are divisible by two, 333 are divisible by 3 , and 199 are divisible by 5 . There are 166 multiples of 6,99 multiples of 10 , and 66 multiples of 15 . And there are 33 numbers that are divisible by 30 . So by the Inclusion-Exclusion Principle there are
numbers that are divisible by at least one of 2,3 , or 5 . Of the remaining numbers, 165 are primes other than 2,3 , or 5 . Note that 1 is neither prime nor composite. This leaves exactly prime-looking numbers.
The problems on this page are the property of the MAA's American Mathematics Competitions