Question

How many two-digit numbers are divisible by 3? [CBSE 2012]

Answer 1

The two-digit numbers divisible by 3 are 12, 15, 18, ..., 99.

Clearly, these number are in AP.

Here, a = 12 and d = 15 − 12 = 3

Let this AP contains n terms. Then,

$$\begin{gathered} a_n=99 \\ \Rightarrow 12+\left(n-1\right)\times 3=99\left[a_n=a+\left(n-1\right)d\right] \\ \Rightarrow 3n+9=99 \\ \Rightarrow 3n=99-9=90 \end{gathered}$$
$\Rightarrow n=30$

Hence, there are 30 two-digit numbers divisible by 3.