Question

How many three-digit numbers are divisible by 9? [CBSE 2013]

Answer 1

The three-digit numbers divisible by 9 are 108, 117, 126, ..., 999.

Clearly, these number are in AP.

Here, a = 108 and d = 117 − 108 = 9

Let this AP contains n terms. Then,

$$\begin{gathered} a_n=999 \\ \Rightarrow 108+\left(n-1\right)\times 9=999\left[a_n=a+\left(n-1\right)d\right] \\ \Rightarrow 9n+99=999 \\ \Rightarrow 9n=999-99=900 \\ \Rightarrow n=100 \end{gathered}$$

Hence, there are 100 three-digit numbers divisible by 9.