How many numbers between 1 to 1000 is divisible by 2 or 5?

Answer 1

Interesting questions, here's how I look at it... Another way of asking the questions is how many multiples of 2 or 5 are there between 1 and 1000. Well if you look at '2' you can start counting the number of terms that 2 is multiplied by another number and their result is less then 1000. For instance, 2*1=2 (so that is one), 2*2=4 (so that is two), 2*3=6 (so that is three) and so on until you get to 2*500=1000. So any number greater then 500 would be larger than 1000. Now, notice that there would be 500 different numbers you could multiple 2 by up to 1000. So there must be 500 terms that are divisible by 2. We repeat this for 5: 5*1=5 (there is the first), 5*2=10 (there is the second)... 5*200=1000. Using the same method as before, you can conclude there are 200 terms that are divisible by 5. Here is where it gets interesting; it would be easy to just say well 500+200=700 terms. But this would not be correct, because each number can only be counted once so terms like 10, 20, 30 were counted for both 2 and 5. So how many terms were double counted, we fortunately we can use the same method as above realizing that this occurred for all number which are a multiple of the product of 2 and 5--so multiples of 10. How many multiples of 10 are there between 1 and 1000? Well, there are 100. So, how many numbers between 1 and 1000 are divisible by 2 or 5? It should be the number of terms divisible by 2 plus the number of terms divisible by 5 minus the number of duplicate terms. 500+200-100 = 600