Final answer:
To find the number of positive integers less than 1000 that are divisible by neither 2, 3, nor 5, we can use the principle of inclusion-exclusion. The total count is 1067.
Step-by-step explanation:
To find the number of positive integers less than 1000 that are divisible by neither 2, 3, nor 5, we can use the principle of inclusion-exclusion. First, let's find the number of positive integers less than 1000 that are divisible by 2, 3, or 5 individually.
The number of integers divisible by 2 is 999/2 = 499.5, so we round down to 499.
The number of integers divisible by 3 is 999/3 = 333.
The number of integers divisible by 5 is 999/5 = 199.8, so we round down to 199.
To find the number of integers divisible by both 2 and 3, we divide 999 by their least common multiple, which is 6. 999/6 = 166.5, so we round down to 166.
To find the number of integers divisible by both 2 and 5, we divide 999 by their least common multiple, which is 10. 999/10 = 99.9, so we round down to 99.
To find the number of integers divisible by both 3 and 5, we divide 999 by their least common multiple, which is 15. 999/15 = 66.6, so we round down to 66.
To find the number of integers divisible by all three (2, 3, and 5), we divide 999 by their least common multiple, which is 30. 999/30 = 33.3, so we round down to 33.
Using the principle of inclusion-exclusion, we can subtract the sum of these counts from the total number of positive integers less than 1000, which is 999.
999 - 499 - 333 - 199 + 166 + 99 + 66 - 33 = 1068.
However, we have overcounted the integers greater than or equal to 1000 that are divisible by 2, 3, or 5. There are 0 integers divisible by 2, 3, or 5 in this range. So we subtract 1 from the total count.
1068 - 1 = 1067.
Therefore, there are 1067 positive integers less than 1000 that are divisible by neither 2, 3, nor 5.