Posted by nancy on Sunday, September 16, 2012 at 12:59pm.
There are 900 three-digit numbers in total (100 to 999 inclusive), and every odd-numbered one is even, i.e. 1st = 100, 3rd = 102, 5th = 104, 7th = 106 etc. There are as many odds and evens in 900 consecutive integers, so exactly half of them are even, i.e. 900/2 = 450.
For part 2, consider just the first 10 integers, i.e. 100 to 109. Of these, 101, 103, 105, 107 and 109 are not even, and of these, only the third one is divisible by 5. So 4 out of the first 10 satisfy the stated conditions for inclusion. Now consider the next 10, i.e. 110 to 119. For any N, if N is odd then (N+10) is also odd, and if N does not divide by 5 then (N+10) does not divide by 5. So exactly the same logic applies to these next ten as to the previous 10. Ditto the next 10, ditto the next 10 after that, etc etc. So the total number between 100 and 999 should be 4 x 900 / 10 = 360.