How many positive integers up to 100 can be divided evenly by either 4 or 6?

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

Step 1: Count integers divisible by 4

The integers divisible by 4 up to 100 can be found by dividing 100 by 4:

  • 100 ÷ 4 = 25

Thus, there are 25 integers that are divisible by 4.

Step 2: Count integers divisible by 6

Next, we count the integers divisible by 6 up to 100 in a similar manner:

  • 100 ÷ 6 = 16 (rounded down)

So, there are 16 integers that are divisible by 6.

Step 3: Count integers divisible by both 4 and 6 (i.e., by 12)

Since some numbers are counted in both previous categories, we need to account for these by finding the numbers divisible by both 4 and 6, which is the least common multiple (LCM) of 4 and 6, i.e. 12:

  • 100 ÷ 12 = 8 (rounded down)

This means there are 8 integers divisible by both 4 and 6.

Step 4: Apply the principle of inclusion-exclusion

The total number of integers divisible by 4 or 6 is calculated as follows:

  • Count(4) + Count(6) – Count(12)
  • 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.

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