Let’s agree to say that a positive integer is prime-like if it is not divisible by 2, 3, or 5. How
many prime-like positive integers are there less than 100? less than 1000? A positive integer
is very prime-like if it is not divisible by any prime less than 15. How many very primelike
positive integers are there less than 90000? Without giving an exact answer, can you
say approximately how many very prime-like positive integers are less than 1010? less than
10100? Explain your reasoning as carefully as you can.
To find the number of prime-like positive integers, we need to understand the divisibility rules of 2, 3, and 5.
A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8). Since we are looking for numbers that are not divisible by 2, we should exclude those that end with an even digit.
A number is divisible by 3 if the sum of its digits is divisible by 3. So, in this case, we need to exclude numbers whose digit sum is divisible by 3.
Finally, a number is divisible by 5 if its last digit is either 0 or 5. Therefore, we should exclude numbers ending with 0 or 5.
Now, let's find the number of prime-like positive integers less than 100:
1. First, let's find all the positive integers less than 100.
We have 99 positive integers: 1, 2, 3, ..., 99.
2. Exclude all numbers divisible by 2:
We know that half of the numbers are divisible by 2 (every other number). So, we need to remove 49 numbers (including 2) to eliminate the even numbers.
3. Exclude all numbers divisible by 3:
To find the numbers divisible by 3, we need to sum the digits of each number. After checking, we find that the numbers 3, 6, 9, 12, ..., 99 are divisible by 3. This gives us 33 numbers to remove.
4. Exclude all numbers divisible by 5:
Out of the remaining numbers, we need to remove those ending with 0 or 5. There are 9 such numbers: 10, 15, 20, ..., 95.
So, the number of prime-like positive integers less than 100 is 99 - 49 - 33 - 9 = 8.
Now, let's find the number of prime-like positive integers less than 1000:
Following the same steps as before:
- The total number of positive integers is 999.
- Excluding numbers divisible by 2 leaves us with 499 numbers.
- Excluding numbers divisible by 3 gives 334 numbers.
- Excluding numbers divisible by 5 gives 93 numbers.
Therefore, the number of prime-like positive integers less than 1000 is 999 - 499 - 334 - 93 = 73.
For the next part, we are looking for very prime-like positive integers less than 90000 (not divisible by any prime less than 15).
We need to find the primes less than 15: 2, 3, 5, 7, 11, 13.
Following similar steps as before, we know that approximately 1/2 of the numbers are divisible by 2, 1/3 by 3, 1/5 by 5, 1/7 by 7, 1/11 by 11, and 1/13 by 13.
Using these fractions, we can estimate the number of very prime-like positive integers by multiplying the total number of positive integers by the product of (1 - (1/2))(1 - (1/3))(1 - (1/5))(1 - (1/7))(1 - (1/11))(1 - (1/13)).
For example, to estimate the number of very prime-like positive integers less than 1010:
- The total number of positive integers is 1010.
- Multiply this by (1 - (1/2))(1 - (1/3))(1 - (1/5))(1 - (1/7))(1 - (1/11))(1 - (1/13)).
Similarly, for estimating the number of very prime-like positive integers less than 10100:
- The total number of positive integers is 10100.
- Multiply this by (1 - (1/2))(1 - (1/3))(1 - (1/5))(1 - (1/7))(1 - (1/11))(1 - (1/13)).
By following this approach, you can estimate the approximate number of very prime-like positive integers.