Question

How many numbers are there between 101 and 999, which are divisible by both 2 and 5? [CBSE 2014]

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,

$$\begin{gathered} a_n=990 \\ \Rightarrow 110+\left(n-1\right)\times 10=990\left[a_n=a+\left(n-1\right)d\right] \\ \Rightarrow 10n+100=990 \\ \Rightarrow 10n=990-100=890 \\ \Rightarrow n=89 \end{gathered}$$

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