Question 1209362: There are 830 composite numbers less than 1000. Let S be the set of composite numbers smaller than 1000 that are not divisible by 2, 3, or 7. How many elements does S have?
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Answer: 120
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Explanation
To find the solution, we'll need the floor function.
This function rounds a positive decimal number down to the nearest integer.
Think of it like a vertical number line.
As the name "floor" implies, we move down to the floor to magnetically lock onto the nearest integer.
Examples:
floor(2.1) = 2
floor(7.9999999) = 7
It might be tempting to round 7.9999999 to 8, but just remember we round down to the floor.
It doesn't matter how close to the ceiling the input is.
Basically the floor function chops off the decimal when looking at positive numbers only.
If the input is an integer, then no changes are made.
There are 999 values in the set {1,2,3,...,999}- Floor(999/2) = 499 of those values are multiples of 2.
- Floor(999/3) = 333 of those values are multiples of 3.
- Floor(999/7) = 142 of those values are multiples of 7.
- Floor(999/(2*3)) = 166 of those values are multiples of 2*3 aka 6.
- Floor(999/(2*7)) = 71 of those values are multiples of 2*7 aka 14.
- Floor(999/(3*7)) = 47 of those values are multiples of 3*7 aka 21.
- Floor(999/(2*3*7)) = 23 of those values are multiples of 2*3*7 aka 42.
Use the inclusion-exclusion principle to say,
499+333+142-166-71-47+23 = 713
A Venn Diagram may come in handy.
If we define P as the set of positive integers smaller than 1000 that are multiples of 2, 3, or 7, then there are 713 values in set P.
Here are the first few items of set P
2,3,4,6,7,8,9,10,12,14,15,16,18,20,21,22
There are 3 prime numbers in set P. Those values are 2,3,7
The remaining 713-3 = 710 items are composite since they are multiples of those trio of values mentioned.
The instructions tell us there are 830 composite numbers in the set {1,2,3,4,...,999}.
Note that 1 isn't composite and it's not prime either.
So there are 830-710 = 120 values in set S.
I have verified the answer is correct using a Python script.
You can take this route as well, or you can use a spreadsheet to verify the answer.
If you aren't given the 830 figure, then you can consult a list of primes to determine there are 168 primes in the set {2,3,...,999}. So that means there are 998-168 = 830 composite numbers in the set {2,3,...,999}. I've excluded the value "1" since it's neither prime nor composite.
The first few values in set S would be 25,55,65,85,95,115,121,125
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