How many positive integers less than 100 are neithe multiples of 2 or 3.
To find the number of positive integers less than 100 that are neither multiples of 2 nor multiples of 3, we can use the principle of inclusion-exclusion.
First, let's find the total number of positive integers less than 100, which is simply 99 (since 100 is not included).
Now, let's find the number of multiples of 2 less than 100. We can do this by dividing 100 by 2:
Number of multiples of 2 = 100 / 2 = 50
Next, let's find the number of multiples of 3 less than 100. We can do this by dividing 100 by 3:
Number of multiples of 3 = 100 / 3 ≈ 33.33
Since we're looking for integers, we'll consider 33 multiples of 3.
Now, we'll use the principle of inclusion-exclusion to find the total count of integers that are neither multiples of 2 nor multiples of 3.
Total = Total integers less than 100 - (Multiples of 2 + Multiples of 3) + (Multiples of 2 and 3)
Total = 99 - (50 + 33) + (Multiples of 6)
Now, let's find the multiples of 6 less than 100:
Number of multiples of 6 = 100 / 6 ≈ 16.67
Considering integers, we'll consider 16 multiples of 6.
Now, we can calculate the total:
Total = 99 - (50 + 33) + 16 = 99 - 83 + 16 = 33
So, there are 33 positive integers less than 100 that are neither multiples of 2 nor multiples of 3.
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