Question

How many different integers between 100 and 500 are multiples of either 6, 8, or both?

Answer 1

101

Explanation:

First, to avoid overcounting, we can make a Venn diagram. The overlap consists of multiples of lcm[6,8], which is 24. The smallest such multiple between 100 and 500 is 5 ∙ 24 = 120, and the largest is 20 ∙ 24 = 480. Since there are 16 integers from 5 through 20 (inclusive), there are 16 multiples of 24 between 100 and 500. Now we complete our diagram so that there are a total of 67 multiples of 6 and 50 multiples of 8, including the overlap. The total number of integers between 100 and 500 that are multiples of 6, 8, or both is 51 + 16 + 34 = 101.

Answer 2

There are (500-100)/2 = 200 numbers divisible by 2 between 100 and 500 counting