The answer is 66. Here's how we find it:
Divisibility Calculation
We need to determine how many positive integers less than 100 are divisible by either 2 or 3. We can do this by finding the number of integers divisible by 2, the number divisible by 3, and then subtracting the number divisible by both (i.e., divisible by 6) to avoid double-counting.
Integers Divisible by 2
According to the provided reference, there are 50 numbers between 1 and 100 that are divisible by 2.
Integers Divisible by 3
The reference also states that 100 divided by 3 is 33.333333, which means there are 33 numbers between 1 and 100 that are divisible by 3.
Integers Divisible by 6
We now need to determine how many integers are divisible by both 2 and 3. A number divisible by both 2 and 3 is divisible by their least common multiple, which is 6. To find the number of integers less than 100 divisible by 6, we calculate 99 / 6 = 16.5. Therefore, there are 16 integers less than 100 that are divisible by 6.
Applying the Inclusion-Exclusion Principle
To find the number of integers divisible by 2 or 3, we use the inclusion-exclusion principle:
- Number divisible by 2 or 3 = (Number divisible by 2) + (Number divisible by 3) - (Number divisible by 6)
- Number divisible by 2 or 3 = 50 + 33 - 16 = 67
Considering numbers less than 100
Since the question specified positive integers less than 100, we are looking at integers from 1 to 99, inclusive. Since our calculations used 100 as the upper limit, it doesn't change the number of integers divisible by 2,3, and 6 within our specified range (1 to 99). If the upper limit were, say, 102, it would impact the number of integers divisible by 2 and 3 and 6. Therefore, our previous calculation holds. The total number divisible by 2 or 3 is 67.
Final Result
Therefore, there are 67 positive integers less than 100 that are divisible by 2 or 3.
