An integer greater than zero is "smiling", if it is certainly not divisible by 2, 3, or 5. How
many such integers are less than 100? are less than 1000?
A positive integer is "extremely smiley" if it'snot divisible by any prime less than 15. Between the numbers 1 and 90000, how many "extremely smiley" positive integers are there?
Without giving an exact answer, approximate the number of "extremely smiley" positive integers that are less than 1010? less than 10100? Clearly explain your reasoning.
many such integers are less than 100? are less than 1000?
A positive integer is "extremely smiley" if it'snot divisible by any prime less than 15. Between the numbers 1 and 90000, how many "extremely smiley" positive integers are there?
Without giving an exact answer, approximate the number of "extremely smiley" positive integers that are less than 1010? less than 10100? Clearly explain your reasoning.
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You could make this complicated by using inclusion-exclusion principle, but I opted for a programming solution using Maple:
c := 0;
for i from 1 to 100 do
if i mod 2 = 0 or i mod 3 = 0 or i mod 5 = 0 then else c := c+1;
end if;
end do;
c;
#1. There are 26 such integers less than 100.
There are 266 such integers less than 1000.
#2. There are 17261 such integers less than 90000.
#3. Based on my method it would be much easier to find an exact solution than to estimate. If you had a different approach, then estimation might make sense. There are 193 extremely smiley positive integers less than 1010 and 1938 less than 10100.
c := 0;
for i from 1 to 100 do
if i mod 2 = 0 or i mod 3 = 0 or i mod 5 = 0 then else c := c+1;
end if;
end do;
c;
#1. There are 26 such integers less than 100.
There are 266 such integers less than 1000.
#2. There are 17261 such integers less than 90000.
#3. Based on my method it would be much easier to find an exact solution than to estimate. If you had a different approach, then estimation might make sense. There are 193 extremely smiley positive integers less than 1010 and 1938 less than 10100.