Final answer:
There are 34 positive integers less than 100 that are neither multiples of 2 nor multiples of 3, calculated by using the principle of inclusion-exclusion.
Step-by-step explanation:
To find how many positive integers less than 100 are neither multiples of 2 nor multiples of 3, we can use the principle of inclusion-exclusion. First, we'll count the number of multiples of 2 and 3, then adjust for the overlap of numbers that are multiples of both.
There are 49 multiples of 2 between 1 and 99 (because 98 is the largest even number less than 100). There are 33 multiples of 3 between 1 and 99 (because 99 is the largest multiple of 3 less than 100). To find the overlap, we need to count the multiples of 6, since every multiple of 6 is also a multiple of both 2 and 3.
There are 16 multiples of 6 less than 100 (since 96 is the largest multiple of 6 less than 100). Now, we subtract the number of multiples of 2 and 3 from the total number of positive integers less than 100, then add back the overlap.
Number of integers neither multiples of 2 nor 3: 100 - 49 - 33 + 16 = 34.
Therefore, there are 34 positive integers less than 100 that are neither multiples of 2 nor multiples of 3.