How many integers from 1 to 1000, inclusive, are divisible by neither 2 nor 5?

Question

How many integers from 1 to 1000, inclusive, are divisible by neither 2 nor 5? Notice the words "neither and nor" in the problem. 

Answer 1

I will be using a trick very similar to what you would see in probability class:

 

The number of integers divisible by 2 from 1 to 1000 is 1000/2 = 500.

The number of integers divisible by 5 from 1 to 1000 is 1000/5 = 200.

The number of integers divisible by 2 AND 5 from 1 to 1000 is 1000/lcm(2,5) = 1000/10 = 100.

 

The number of integers divisible by 2 OR 5 from 1 to 1000 is 500 + 200 - 100 = 600. <-- union minus intersection from probability

 

The number of integers divisible by NEITHER 2 NOR 5 from 1 to 1000 is 1000 - 600 = 400. <-- complement from probability

 

So your answer is 400 integers.

 

These integers are the numbers ending in 1, 3, 7, and 9 (just in case you are curious).

Answer 2

If not divisible by 2, they're odd.  If not divisible by 5, they don't end in 5 (numbers such as 10 and 20 are already eliminated by the elimination of all evens).

 

Now think about all of the numbers from 1 to 1000 that remain after these deletions.

 

To get a handle on this problem, you might want to first apply this criterion to just the numbers from 1 to 100 (this is the idea of scaling down a hard problem on the first analysis of it).