Question

How many two-digit numbers are divisible by $6$?

Answer 1

Step 1. List of numbers:

Two digit numbers which are divisible by $6$ is $12,18,24,30,36,42,\dots ,96.$
Here, $18-12=24-18=6$
So, the list of numbers $12,18,24,30,36,42,\dots ,96$ forms an AP, whose first term ($a$) = $18$, and common difference ($d$) = $6$.

Step 2. Finding the value of $n$:

Let there are $n$ terms in the above sequence, then
$a_n=96$
The formula for Arithmetic Progression AP is
$\begin{array}{cc}\Rightarrow & a+\left(n-1\right)d=a_n\\ \Rightarrow & a+(n-1)d=96\\ \Rightarrow & 12+(n-1)6=96\\ \Rightarrow & 12+6n-6=96\\ \Rightarrow & 6+6n=96\\ \Rightarrow & 6n=96-6\\ \Rightarrow & 6n=90\\ \Rightarrow & n=\frac{90}{6}\end{array}$

Therefore, the value of $nis15$.