Yes, 74 is the correct answer. Let me expand on what @Lightlyseared wrote,
We want to pair each factor of 5 with a factor of 2. It should be apparent that there are many more 2’s than 5’s, so we are limited by the number of 5’s. We can just count 5’s, knowing there will be a sufficient number of 2’s to pair with.
Every fifth number is divisible by 5, so there are 300/5=60 numbers with at least one factor of 5. To find how many additional 5’s we get from numbers divisible by 25, we can divide 300 by 25 or, more simply, divide 60 by 5, to get 12. For numbers that contribute yet another 5, we can divide 300 by 125 or 12 by 5 to get the 2 numbers 125 and 250. 60 + 12 + 2= 74.