How many integers in the range 1 through 120 are integer multiples of 2, 3, or 5?

There are 88.

There are 60 multiples of 2.  We can arrive at these by counting the numbers divisible by 2:  2, 4, 6, 8, and 0.  5 of these * 12 sets of 10 = 60 multiples.

After taking these out, there are 20 multiples of 3 left.  3 and 9 are the only digits left that we have not accounted for that are divisible by 3; 2*10 = 20 multiples.

After taking these out, we have 8 multiples of 5 that have not yet been included.  For this, I found it easier to just count by 5s and only count the ones that were not even or divisible by 3.

60+20+8 = 88.