Counting problem

The smallest multiple of 5 is \(100 \Rightarrow (20 \times 5)\), and the largest multiple of 5 is \(995 \Rightarrow (199 \times 5)\). There are \(199 - 20 + 1 = 180\) multiples of 5

 

The smallest multiple of 8 is \(104 \Rightarrow (13 \times 8)\), and the largest multiple of 5 is \(995 \Rightarrow (124 \times 8)\). There are \(124 - 13 + 1 = 122\) multiples of 8

 

However, we overcount every \(\text{lcm}(5,8) = 40\) numbers.

 

The smallest number we overcount for is \(120 \Rightarrow (3 \times 40)\), and the largest number is \(960 \Rightarrow (24 \times 40)\). There are \(24 - 3 + 1 = 22\) numbers we overcount for. 

 

Can you take it from here?

There are: 900 / 5 ==180 multiples of 5

 

There are: 900 / 8 ==112 multiples of 8

 

There are: 900 / 40 ==22 multiples of 40

 

900 - 180 - 112 + 22 ==630 integers that are neither multiples of 5 or 8