Question

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

Answer 1

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,

$a_n=990$

$\Rightarrow 110+(n-1)\times 10=990$

$\Rightarrow 10n+100=990$

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

$\Rightarrow n=89$

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