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 The final quantity (Posted on 2015-02-18)
Given a list of all composite numbers below 1000, how many will remain after erasure of numbers divisible by 2,3 or 5?

Rem1: "or" is inclusive i.e. and/or.
Rem2: number 1 is neither prime nor composite, so it does not appear on the initial list.

 No Solution Yet Submitted by Ady TZIDON No Rating

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 A third method by hand Comment 4 of 4 |
The LCM (Least Common Multiple) or 2,3, and 5 is 30.  There are 8 numbers between 1 and 30 that are not divisible by 2, 3 or 5.  There are 2 numbers between 1 and 9 that are not divisible by 2, 3 or 5.

999 = 33*30 + 9, so there are 33*8 + 2 = 266 numbers less than 1000 that are not divisible by 2, 3 or 5.

But these include primes.  There are 168 primes under 1000, but these include 2,3,and 5.  There are therefore 165 primes under 1000 that are not divisible by 2,3,and 5.

266 - 165 = 101, so there are 101 non-primes under 1000 that are not divisible by 2,3, or 5.  One of these (1) is neither prime nor composite, and needs to be subtracted out also.

 Posted by Steve Herman on 2015-02-19 22:15:37

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