How many numbers are there between 101 and 999, which are divisible by both 2 and 5?

The numbers which are divisible by both 2 and 5 are divisible by 10 also.

Now, the numbers between 101 and 999 which are divisible 10 are 110, 120, 130, ..., 990.

Clearly, these number are in AP.

Here, a = 110 and d = 120 − 110 = 10

Let this AP contains n terms. Then,

$a_{n}=990$

$\Rightarrow 110+(n-1) \times 10=990 \quad\left[a_{n}=a+(n-1) d\right]$

$\Rightarrow 10 n+100=990$

$\Rightarrow 10 n=990-100=890$

$\Rightarrow n=89$

Hence, there are 89 numbers between 101 and 999 which are divisible by both 2 and 5.