Question

How many numbers between 1 and 2005 are integer multiples of 3 or 4 but not 12?

Answer 1

Count the number of multiples of 3, 4, and 12 in the range 1-2005:

⌊2005/3⌋ ≈ ⌊668.333⌋ = 668

⌊2005/4⌋ = ⌊501.25⌋ = 501

⌊2005/12⌋ ≈ ⌊167.083⌋ = 167

(⌊x⌋ means the "floor" of x, i.e. the largest integer smaller than x, so ⌊a/b⌋ is what you get when you divide a by b and ignore the remainder)

Then using the inclusion/exclusion principle, there are

668 + 501 - 2•167 = 835

numbers that are multiples of 3 or 4 but not 12. We subtract the number multiples of 12 twice because the sets of multiples of 3 and 4 both contain multiples of 12. Subtracting once removes the multiples of 3 and 4 that occur twice. Subtracting again removes them altogether.