We say that a positive integer is quiteprime if it is not divisible by 2, 3, or 5. How many quiteprime positive integers are there less than 100? less than 1000? A positive integer is very quiteprime if it is not divisible by any prime less than 15. How many very quiteprime positive integers are there less than 90000? Without giving an exact answer, can you say approximately how many very quiteprime positive integers are less than 1010? less than 10100? Explain your reasoning as carefully as you can.
I will guide you on how to solve it for the first two, that is your "quiteprimes" under 100 and under 1000.
1) Numbers under 100:
Since you are not allowed to divide by primes LESS THAN 2, 3, and 5, that means you must consider prime factors of 7 and higher. As a consequence, you must divide 100 / 7 =floor(14). Then you must count ALL prime number between 7 and 14. And you have:7, 11 and 13 =3. Next:floor(100/11)=9 and 9 < 11, therefore there are no prime to consider.
So, the total number of "quiteprimes" is the number of prime numbers as we calculated them above =3
2)Numbers under 1000
This is very similar to above calculation for 100, except for the larger number of 1000:
You will divide: 1000 / 7=floor(142). Then you will have to count all prime number between 7 and 142 and you should get: 31. Next you will divide 1000 / 11 =floor(90). Then you will count all prime numbers between 11 and 90 and you should get:20. Then you would continue with this process for: 1000 / 13 =floor(76) and count all the prime numbers beteen 13 and 76 and you should get:16.......and so on. Then you would add them all up:31 + 20 + 16.......etc. and you should get: 94.
3) All other examples in your question are calculated in EXACTLY the same way, except for a big number like 90,000 it will take quite a bit of work to count them all.
Note: There maybe an easier way of calculating them, but I don't know of any. Good luck to you.