Final answer:
There are 33 numbers among the first 100 positive integers that are not multiples of two and three, after accounting for every number and removing those that are divisible by either 2, 3, or both.
Step-by-step explanation:
To determine how many of the first 100 positive integers are not multiples of two and three, we must exclude all the numbers that are divisible by 2 (even numbers) and all the numbers that are divisible by 3.
To find the numbers not divisible by either 2 or 3, we can take a two-step approach:
Count the number of multiples of 2 among the first 100 positive integers, which are even numbers. There are 100/2 = 50 such numbers.
Count the number of multiples of 3 among the first 100 positive integers. There are 100/3 = 33 such numbers, but since some of these are also multiples of 2 (such as 6, 12, 18, ...), we only count each of those multiples once.
To find the common multiples of both 2 and 3, which are multiples of 6 (since 2 x 3 = 6), we count these among the first 100 positive integers. There are 100/6 = 16 such numbers.
We then calculate the total number of non-multiples by subtracting the individual counts and adding back the count of common multiples (since they were subtracted twice) from the total of 100:
Total non-multiples = 100 - 50 (multiples of 2) - 33 (multiples of 3) + 16 (common multiples of 2 and 3)
After calculating, we find there are 33 non-multiples of both 2 and 3 within the first 100 positive integers.