How many positive integers not exceeding 100 are divisible either by 4 or by 6?

To find out how many positive integers not exceeding 100 are divisible by either 4 or 6, we can use the principle of inclusion-exclusion.

First, let’s determine how many numbers are divisible by 4:

• The largest integer divisible by 4 that is less than or equal to 100 is 100 itself (100 ÷ 4 = 25).
• Thus, there are 25 numbers (4, 8, 12, …, 100).

Next, we find how many numbers are divisible by 6:

• The largest integer divisible by 6 that is less than or equal to 100 is 96 (100 ÷ 6 = 16).
• Thus, there are 16 numbers (6, 12, 18, …, 96).

Now we must consider those numbers that are counted in both sets, specifically those divisible by both 4 and 6 (i.e., by 12, their least common multiple):

• The largest integer divisible by 12 that is less than or equal to 100 is 96 (100 ÷ 12 = 8).
• Thus, there are 8 numbers (12, 24, 36, …, 96).

Now we apply the inclusion-exclusion principle:

Count = (Count of multiples of 4) + (Count of multiples of 6) - (Count of multiples of 12)

Inserting our findings, we get:

Count = 25 + 16 - 8 = 33

Therefore, the total number of positive integers not exceeding 100 that are divisible either by 4 or by 6 is 33.