Question

How many two digit numbers are divisible by $3$ ?

Answer 1

Step 1: Identify the sequence as an arithmetic progression

The first two digit number divisible by $3$ is $12$

The last two digit number divisible by $3$ is $99$

The sequence of numbers $12,15,18,....,99$ which are divisible by $3$ is an arithmetic progression with $1^{st}$ term $12$ and common difference of $3$

Step 2: Find the number of terms in an arithmetic progression

The $n^{th}$ term of an arithmetic progression is given as

$a_n=a+(n-1)d$

Here,$a_n=99,a=12,d=3$

$$\begin{gathered} \Rightarrow 99=12+(n-1)3 \\ \Rightarrow 87=3n-3 \\ \Rightarrow 3n=90 \\ \Rightarrow n=30 \end{gathered}$$

Hence, the number of two digit numbers divisible by $3$ is $30$