How many positive integers less than 500 are divisible by 6, 8 or 10?

Let m(a), m(a & b), m(a or b) denote the number of multiples of a, a and b, and a or b, respectively, that are less than 500. We can use the formula:

m(6 or 8 or 10) = m(6) + m(8) + m(10) - m(6 & 8) - m(6 & 10) - m(8 & 10) + m(6 & 8 & 10)

We have:

m(6) = (498 - 6) / 6 + 1 = 83


m(8) = (496 - 8) / 8 + 1 = 62


m(10) = (490 - 10) / 10 + 1 = 49

m(6 & 8) = m(24) = (480 - 24)/24 + 1 = 20

m(6 & 10) = m(30) = (480 - 30)/30 + 1 = 16

m(8 & 10) = m(40) = (480 - 40)/40 + 1 = 12

m(6 & 8 & 10) = m(120) = (480 - 120) / 120 + 1 = 4

Therefore,

m(6 or 8 or 10) = 83 + 62 + 49 - 20 - 16 - 12 + 4 = 194 - 48 + 4 = 150

Answer: 150

(Note: the official answer is D: 151. However, the problem should be stated as “less than or equal to 500” instead of “less than 500.” If that is the case, m(10) = 50, instead of 49.)